The maximum diameter of $d$-dimensional simplicial complexes

The maxim um diameter of d -dimensional simplicial complexes Stefan Glo c k ∗ Olaf P arczyk †‡ Silas Rathk e † Tib or Szab ó † Abstract F or ev ery ﬁxed dimension d and suﬃciently large n , we determine the maxim um possible diameter of a strongly connected d -dimensional simplicial complex on n v ertices. This impro ves on a sequence of previous results and settles a problem of Santos from 2013. On the wa y , as a sp ecial case, we also c haracterise the existence of an extra-tigh t Euler tour in the complete d -uniform h yp ergraph on n v ertices. 1 In tro duction The Polynomial Hirsch Conjecture is a central problem of Discrete Geometry , stating that the diameter of the v ertex/edge gr aph of a p olytop e is at most a p olynomial in the n umber of its facets and the dimension. While this conjecture is certainly captiv ating for its own sake, its broader importance stems from its direct connection to one of the most fascinating ma jor op en problems of Discrete Optimization: the p olynomialit y of the Simplex Metho d. Should the conjecture turn out to b e false, then no pivo t rule can b e p olynomial in the w orst case. (The original conjecture of Hirsc h from 1957, stating the speciﬁc upp er bound of n − d , w as famously dispro ved b y San tos [ 21 ] in 2010.) Due to the apparent diﬃcult y of understanding the ﬁne geometry of p olytop es, most approac hes tow ards b ounding the diameter rely on combinatorial abstractions. These strip aw ay muc h of the geometry and focus solely on well-c hosen com binatorial prop erties of the set of facets. The classic quasi-p olynomial upp er b ound of Kalai and Kleitman [ 17 ], whic h is the basis of the subsequent b est known b ounds (T odd [ 24 ], Sukega w a [ 23 ]), also turns out to b e of this t yp e. The com binatorial b ase abstr actions of Eisenbrand, Hähnle, Razb oro v, and Roth v oß [ 10 ] generalize man y of the kno wn abstractions in to the same simple framework. The P olymath3 pro ject [ 16 ] initiated b y Gil Kalai in 2009 was sp eciﬁcally targeted to wards identifying and making progress on v arious combinatorial abstractions. In this pap er, we consider the diameter problem for abstract simplicial complexes, a natural abstrac- tion studied by San tos, and provide the precise answer. Our inv estigations lead us to deﬁne a certain highly regular, global com binatorial structure and we characterise its existence in the complete uniform h yp ergraph. A simplicial c omplex on n vertices is a family C of subsets of [ n ] which is closed under taking subsets. The maximal elements of C are called fac ets . W e call C a (simplicial) d -c omplex , if each of its facets has size d + 1 . The dual gr aph G ( C ) of a simplicial d -complex C is deﬁned on the set F ( C ) of facets as its vertex set, with tw o facets forming an edge in G ( C ) if their intersection has size d . (Sets of size d are also called ridges and the dual graph is sometimes referred to as the facet-ridge graph.) W e say C is str ongly c onne cte d if G ( C ) is connected. The diameter of C is the diameter of its dual graph G ( C ) (i.e., the maxim um, taken ov er all pairs u, v ∈ V ( G ( C )) of v ertices, of the length of a shortest u - v -path). San tos [ 22 ] deﬁned H s ( n, d ) to b e the maximum diameter of a strongly connected simplicial d -complex on [ n ] (note that in his notation, d is shifted b y one). W e refer the reader to Section 8 for a description of the connection to the diameter of p olytop es. Santos [ 22 ] pro ved for ﬁxed d ≥ 2 that Ω  n 2 d +2 3  ≤ H s ( n, d ) ≤ 1 d  n d  . ∗ F akultät für Informatik und Mathematik, Universität Passau, Innstraße 41, 94032 Passau, Germany . Email: stefan.glock@uni-passau.de . † F ach b ereic h Mathematik und Informatik, F reie Universität Berlin, Arnimallee 3, 14195 Berlin, German y . Emails: parczyk@mi.fu-berlin.de , szabo@mi.fu-berlin.de , s.rathke@fu-berlin.de . ‡ Department AI in Society , Science, and T ec hnology , Zuse Institute Berlin, T akustraße 7, 14195 Berlin, Germany . 1 F or the upp er b ound, his simple volume argumen t in fact implies H s ( n, d ) ≤  1 d  n d  − d + 1 d  . (1) F or the lo wer b ound, San tos [ 22 ] giv es an explicit pro duct construction. An alternative lo wer b ound of the order n d/ 4 is con tained in a pap er by Kim [ 19 ] (cf. [ 6 ]). Subsequen tly , a series of authors used a v ariet y of diﬀeren t to ols to construct complexes of larger and larger diameter. Criado and Santos [ 6 ] ga ve an explicit algebraic construction of simplicial d -complexes using ﬁnite ﬁelds, whose diameter, Θ( n d ) , matched the order of magnitude of the upp er bound for every ﬁxed d . Criado and Newman [ 5 ] in tro duced probabilistic constructions to the problem and reduced the gap b etw een the upp er and lo wer b ounds, from a factor exp onen tial in d to O ( d 2 ) . Most recently , Bohman and Newman [ 3 ] managed to pin down the precise asymptotics for ev ery ﬁxed d ≥ 2 applying the diﬀerential equations method to track the evolution of a random greedy algorithm to construct the desired d -complex with large diameter:  1 d − (log n ) − ε   n d  ≤ H s ( n, d ) , (2) where ε < 1 /d 2 and n is suﬃcien tly large. W e note (cf. [ 20 ]) here that an earlier result of Dębski, Lonc, and Rzążewski [ 7 ] on harmonious colorings of fragmen table k -uniform hypergraphs also can be used to deriv e an asymptotically precise low er b ound of  1 d − o (1)   n d  on H s ( n, d ) , without an explicit error term. Just very recen tly , Gould and Kelly [ 14 ] improv ed the error term in ( 2 ) from logarithmic to O ( n − 1 /d ) (as an application of a general theorem on certain h yp ergraph matchings). In [ 20 ], the last three authors gav e explicit constructions of simplicial 2 -complexes whose diameter attains the upp er b ound ( 1 ) for all n , except for 6 : H s ( n, 2) = (  1 2  n 2  − 3 2  n  = 6 5 =  1 2  6 2  − 3 2  − 1 n = 6 . (3) In [ 20 ] it was also conjectured that the simple upp er b ound ( 1 ) of San tos can in fact b e ac hieved for all ﬁxed d , as long as n is large enough. Here we prov e this conjecture. Theorem 1.1. F or every p ositive inte ger d ≥ 2 , ther e exists a p ositive inte ger n 0 such that for al l n > n 0 , H s ( n, d ) =  1 d  n d  − d + 1 d  . Our construction of a d -dimensional simplicial complex for Theorem 1.1 starts out with a reduction to a design theoretic problem on almost-complete d -uniform h yp ergraphs. F or this, the concept of “turns”, whic h w as used in [ 20 ] to handle the ad ho c analysis of certain cases of the pro of, will b e instrumental and emplo yed in a far more abstract setting. With them, we will b e able to construct a relatively short simplicial complex whose dual graph is a path and has a giv en mo dular degree sequence. The rest of the simplicial complex can then b e obtained through the existence of a structure w e call extra-tight Euler trails in almost-complete h yp ergraphs. The pro of of this existence theorem (Theorem 1.8 ) takes up a signiﬁcan t p ortion of our pap er. The proliferation of ad ho c complications that arise in some of the 2 -dimensional constructions of [ 20 ] suggests that an explicit approac h in higher dimension is not likely to b e work able. And indeed, our construction for Theorem 1.8 emplo ys the probabilistic technique of absorption. F or its pro of we tak e inspiration from several earlier pap ers [ 11 , 12 , 13 ], and combine concepts and ideas developed there with new ingredients. Remark. As it was noted b y Santos [ 22 ], the maximum diameter H s ( n, d ) is equal to the length of the longest induced path in the Johnson graph J ( n, d + 1) , which is th us also determined in Theorem 1.1 . 1.1 Hyp ergraphs Simplicial d -complexes C are in one-to-one correspondence with ( d + 1) -graphs F where F is the set of facets of C and C is the simplicial d -complex ⟨F ⟩ := { F ′ : ∃ F ∈ F , F ′ ⊆ F } generated b y F . W e deﬁne 2 the dual graph G ( F ) and diameter of a ( d + 1) -graph F to b e those of the d -complex ⟨F ⟩ . The shadow ∂ F of F is the d -graph containing all those d -sets that are subsets of a member of F . Since for an y ( d + 1) -graph G , the dual graph of a subfamily F ⊆ G corresponding to the v ertices of a shortest path in G ( G ) is a path itself, the maximum diameter H s ( n, d ) can alw ays b e attained by a ( d + 1) -family whose dual graph is a path. The ﬁrst part of the next observ ation giv es the simple argumen t for ( 1 ) when the dual graph is a path, whic h thus implies the upp er b ound in Theorem 1.1 . Moreo ver, we also obtain simple conditions, in terms of the shadow, for attaining this bound. Observ ation 1.2. L et F b e a ( d + 1) -gr aph on vertex set [ n ] . (i) If G ( F ) is a p ath, then | ∂ F | = |F | · d +1 . In p articular, the length of the p ath is at most  1 d  n d  − d +1 d  and e quality holds if and only if  n d  − | ∂ F | ≤ d − 1 . (ii) If G ( F ) is a cycle, then | ∂ F | = |F | · d . In p articular, if ∂ F =  [ n ] d  then for any F ∈ F the ( d + 1) -gr aph F ′ = F \ { F } has diameter  1 d  n d  − d +1 d  . 1 4 7 5 2 6 10 7 3 6 9 8 1 5 3 1 3 5 6 2 5 9 6 4 7 3 8 1 9 2 4 1 7 8 Figure 1: On the left: A strongly connected simplicial 2 -complex on [9] with maximum diameter. The facets are giv en by the triangles. T o ensure that the dual graph is a path, eac h pair in  [9] 2  app ears at most once as an edge. The diameter is maximum b y Observ ation 1.2 b ecause only the pair { 4 , 7 } do es not app ear as an edge. On the right: A (non-optimal) straight simplicial 2 -complex on [10] . Pr o of. F or part (i) , we consider the members F 1 , . . . , F |F | of F in the order they app ear on the path of the dual graph. The set F 1 con tains ( d + 1) d -subsets and each subsequen t F i con tains d such d - subsets that were not con tained in any of the previous F j s (otherwise G ( F ) con tains a cycle). The b ound on the length |F | − 1 of the path then follows since ∂ F ⊆  [ n ] d  . Equality holds if and only if 1 d ( | ∂ F | − 1) = |F | =  1 d  n d  − 1  , which is equiv alent to  n d  − ( d − 1) ≤ | ∂ F | . F or part (ii) , we consider the mem b ers of F in an order as they appear on the cycle of the dual graph. The counting of d -subsets is the same as in part (i) un til the very last vertex, which has t wo neigh b ours the d -subsets of which w ere already accoun ted for. So for the last vertex of the cycle we need to coun t only ( d − 1) new d -subsets, hence the formula for the shadow follows. Removing F from F mak es the dual graph into a path and from the shado w ( d − 1) of the d -subsets get remov ed. Now if ∂ F =  [ n ] d  , then | ∂ ( F \ { F } ) | =  n d  − d + 1 so the diameter of F \ { F } is  1 d  n d  − d +1 d  b y part (i) . A sp ecial, v ertex-sequential wa y of constructing a ( d + 1) -family with a path dual graph is via an appropriate sequence T = ( v 1 , . . . , v k ) ∈ [ n ] k of (not necessarily distinct) vertices, where one tak es the sets formed by d + 1 consecutive en tries of T . T o hav e a simplicial d -complex, one should of course assume that no element of [ n ] rep eats within d + 1 consecutive entries of T , but then consecutive ( d + 1) - sets automatically hav e a d -element intersection, so the dual graph c ontains a spanning path (of length k − d − 1 ). W e are interested in the case when there are no more edges in it. Deﬁnition 1.3. F or a sequence T = ( v 1 , . . . , v k ) ∈ [ n ] k , let S T = S ( d +1) T := {{ v i , v i +1 , . . . , v i + d } : i = 1 , . . . , k − d } . If the hypergraph S T is ( d + 1) -uniform and G ( S T ) is a path, then S T , as w ell as the d -complex it generates, is called str aight . 3 The d -complexes of [ 5 , 6 , 3 ], achieving maximum diameter in the order of magnitude, are all of this sp ecial form. The extremal 2 -complexes of [ 20 ] also contain signiﬁcant straight p ortions, but, for most v alues of n , are not en tirely of that type. The situation will be similar for our d -complexes establishing the precise Theorem 1.1 . This is b ecause ( d + 1) -graphs achieving maximum diameter can only b e straigh t for certain v alues of n , those satisfying some divisibility conditions. Y et, as we will see, a straight ( d + 1) - graph of a maxim um diameter is p ossible for an inﬁnite sequence of n , leading us to the deﬁnition and construction of certain highly regular combinatorial structures we call extra-tigh t Euler tours. The k ey prop ert y of straight simplicial complexes, that their dual graph G ( S T ) do es not con tain mor e edges than its canonical spanning path, means that the non-neighbouring ( d + 1) -in terv als of T must hav e at most d − 1 common elements. This is most con venien tly formulated in the realm of d -graphs: every d -subset of [ n ] should o ccur at most once among d + 1 consecutiv e en tries of T . Deﬁnition 1.4. Given a d -graph H , an extr a-tight tr ail T in H is a sequence ( v 1 , . . . , v k ) of ver- tices of H suc h that the sets e ι,σ : = { v ι , . . . , v ι + d }\{ v ι + σ } with ι ∈ [ k − d ] , σ ∈ [ d ] and the set e k − d +1 ,d : = { v k − d +1 , . . . , v k } are all pairwise distinct and in H . The ends of T are ( v 1 , . . . , v d ) and ( v k − d +1 , . . . , v k ) . An extr a-tight tour C in H is a (cyclic) v ertex sequence ( v 1 , . . . , v k ) such that the edges e ι,σ : = { v ι , . . . , v ι + d }\{ v ι + σ } with ι ∈ [ k ] , σ ∈ [ d ] are all pairwise distinct and all in H (where v k + i : = v i for i ∈ [ d ] ). W e sa y that the ab ov e edges e ι,σ are c over e d by the extra-tight trail (respectively , tour). Observ e that a sequence T = ( v 1 , . . . , v k ) ∈ [ n ] k forms an extra-tigh t trail in K ( d ) n if and only if S T is a straight ( d + 1) -graph. Analogously , a (cyclic) sequence T = ( v 1 , . . . , v k ) forms an extra-tight tour in K ( d ) n if and only if the h yp ergraph C T := {{ v i , v i +1 , . . . , v i + d } : i = 1 , . . . , k } (where v k + j : = v j for j ∈ [ d ] ) is a ( d + 1) -graph whose dual graph is a cycle. Remark 1.5. Observ e that for the sequence T of an extra-tigh t trail (resp ectiv ely , tour) in K ( d ) n , the edges co vered by T are exactly the ones contained in the d -graph ∂ S T (resp ectiv ely , ∂ C T ). F or ease of notation, whenev er it do es not cause confusion, we will also use the letter T to denote the d -graph ∂ S T (resp ectiv ely , ∂ C T ) itself. Deﬁnition 1.6. An extra-tight trail (resp ectively , tour) T in a d -graph H whic h cov ers all edges of H is called an extr a-t ight Euler tr ail (resp ectively , extr a-tight Euler tour ) of H . If T is an extra-tigh t Euler tour of K ( d ) n then ∂ C T =  [ n ] d  , so by Observ ation 1.2(ii) deleting an y mem b er of C T pro vides a construction establishing Theorem 1.1 . The name of the concept in the previous deﬁnition is inspired by the generalisation of usual Euler tours in graphs to d -graphs. A (cyclic) sequence T of vertices in a d -graph G is called an Euler tour if every d cyclically consecutive en tries of the sequence forms an edge of G and every edge of G app ears exactly once as such. An Euler tour of the complete d -graph K ( d ) n means a (cyclic) sequence of  n d  v ertices from [ n ] , such that eac h d -subset of [ n ] app ears uniquely as d cyclically consecutiv e entries of the sequence. The existence of this beautiful ob ject of co ding theory was famously conjectured by Chung, Diaconis, and Graham [ 4 ] (they called them universal cycles), whenev er d divides  n − 1 d − 1  . Glo c k, Joos, Kühn, and Osth us [ 12 ] prov ed this in m uch greater generalit y: for d -graphs, that besides satisfying the immediate divisibilit y conditions on v ertex degrees, migh t hav e a minim um ( d − 1) -degree which is a linear fraction less than that of the complete graph. F or the existence of an extra-tight Euler tour in K ( d ) n certain degree conditions are also necessary . Namely , if T is an extra-tigh t Euler tour of a d -graph G , then every o ccurrence of an element a ∈ [ n ] as an entry of T is contained in d 2 edges of T (cf. Observ ation 2.1 ). Since the deﬁnition of an extra-tight Euler tour requires these d -subsets to be distinct for diﬀerent occurrences of a in T and together they should include all d -subsets of G containing a , an extra-tigh t Euler tour can only exist if the degree of a in G is divisible by d 2 . In particular, if an extra-tight Euler tour exists in the clique K ( d ) n , then d 2 divides  n − 1 d − 1  . Our next theorem states that these divisibility conditions are also suﬃcien t for large enough n , ev en if the minim um ( d − 1) -degree is a small linear fraction smaller than that of the clique. Theorem 1.7. F or every inte ger d ≥ 2 , ther e ar e an α > 0 and a p ositive inte ger n 0 such that the fol lowing holds for every inte ger n ≥ n 0 . If G is an n -vertex d -gr aph with δ ( G ) ≥ (1 − α ) n for which al l vertex de gr e es ar e divisible by d 2 , then it has an extr a-tight Euler tour. In p articular, K ( d ) n has an extr a-tight Euler tour if and only if d 2 divides  n − 1 d − 1  . 4 As w e argued ab ov e, this theorem together with Observ ation 1.2 implies Theorem 1.1 whenever d 2 divides  n − 1 d − 1  . This means a p ositiv e fraction of the in tegers n for any d , including for instance every n ≡ 1 (mo d d 2 ) . The correct asymptotics is also implied for ev ery n , with an error term of O ( n − 1 ) , whic h is b etter than that of any of the previous asymptotic results [ 3 , 7 , 14 ]. Y et, from the p oint of view of the precise v alue, this residue class lo oks like the “easiest” of the cases, as the optimal ( d + 1) -graph must miss co vering d − 1 of the d -subsets. A ccording to Observ ation 1.2 , this is the most “wiggle ro om” w e can p ossibly hav e. And indeed, the resolution of the 2 -dimensional case in [ 20 ] should b e a warning sign. There nice, explicit cyclic constructions were found whenev er n − 1 is divisible by 2 2 = 4 . F or the other residue classes, how ev er, more and more in tricate ad ho c adjustmen ts had to b e introduced to ac hieve the optimal diameter. So even if w e had access to relatively simple explicit extra-tight Euler tours for some residue class (which we do not ha ve, not even for d = 3 ), the complications to extend this to all n w ould quite likely b e horrendous/infeasible. Indeed our approach for the pro of of Theorem 1.1 is very diﬀerent. It a voids the separate treatment of diﬀerent residue classes, and rather relies on the existence of very long extra-tigh t Euler trails in h yp ergraphs that are allow ed to be considerably sparser than complete. Our pro of of Theorem 1.7 will also go through this theorem. F or extra-tight trails, the divisibility constraints on the degrees are a bit more complicated for the v ertices at the ends, but again w e can sho w that these, together with a high enough minimum ( d − 1) -degree are suﬃcient. Theorem 1.8. F or every inte ger d ≥ 2 , ther e ar e an α > 0 and a p ositive inte ger n 0 such that the fol lowing holds for every inte ger n ≥ n 0 . L et G b e an n -vertex d -gr aph with δ ( G ) ≥ (1 − α ) n and { v 1 , . . . , v d } , { v ′ 1 , . . . , v ′ d } disjoint e dges in G such that deg G ( v ) ≡ ( i ( d − 1) + 1 (mo d d 2 ) ∃ i ∈ [ d ] : v ∈ { v i , v ′ i } 0 (mo d d 2 ) else. (4) Then ther e exists an extr a-tight Euler tr ail with the ends ( v 1 , . . . , v d ) and ( v ′ d , . . . , v ′ 1 ) . The straight simplicial complex of Bohman and Newman [ 3 ] is constructed through a random se- quence of vertices forming an extra-tight trail and th us provides an asymptotically optimal solution for Theorem 1.8 in K ( d ) n . Their random pro cess how ever gets stuck at some point, in the sense that the addition of an y v ertex as the next entry of the sequence would violate the extra-tight trail conditions. T o a void getting stuck, it is natural to emplo y the metho d of absorption. W e start with constructing a sp ecial extra-tight trail A called “absorb er”, whic h has a lot of built-in ﬂexibility , so that it is able to incorp orate in to itself the leftov er edges L of any asymptotically optimal extra-tight trail our pro cess ends up building. By this we mean that A ∪ L can be co vered by an extra-tight trail whose ends are the same as those of A . W e mak e sure that the asymptotically optimal extra-tight trail our pro cess builds con tains A as a subsequence and hence when the pro cess gets stuck, the leftov er can b e absorb ed in to it, i.e. the subsequence of A is replaced by that of the extra-tight Euler trail of A ∪ L . F or our pro of w e adapt some of the heavy mac hinery developed by the ﬁrst author together with Kühn, Lo, and Osth us [ 13 ] in their w ork on decomp ositions of hypergraphs and combine it with several no vel ideas. One of these con tributions includes in tricate switc her mec hanisms, built into the absorber trail, the role of whic h is to correct the “mistakes” which happ en during the absorption pro cess. Another no velt y is a signiﬁcan t strengthening of the existing asymptotically optimal constructions [ 3 , 7 , 14 ], which also ensures that the maxim um degree of the leftov er hypergraph can b e arbitrarily small. 1.2 Pro of o v erview and organization of the pap er. W e no w outline the ov erall pro of structure which is also illustrated in Figure 2 . In Section 2 , we derive Theorem 1.7 and Theorem 1.1 from Theorem 1.8 . The former is quite straightforw ard; for the latter, one needs to resolv e the issue that the divisibility condition on the vertex degrees might not b e satisﬁed. W e o vercome this obstacle by using th e fact that our simplicial d -complex is not required to b e straight, but can mak e “turns” (cf. Figure 1 ). This will allo w us to build an initial segment of our simplicial complex suc h that remo ving the d -sets they cov er (together with up to d − 1 additional d -sets), we are left with a d -graph satisfying the divisibilit y and minimum degree conditions of Theorem 1.8 . The application of The orem 1.8 then giv es us a straigh t simplicial complex, which can b e concatenated with the initial segmen t, resulting in a simplicial complex cov ering all but at most d − 1 of the d -sets of [ n ] , whic h is 5 Appro ximate Decom- p osition Lemma 3.1 Sup ercomplex Lemma Co ver Down Lemma 3.2 Theorem 1.8 Absorb er Lemma 3.3 V ortex Lemma 3.5 Theorem 1.1 Observ ation 2.3 Theorem 1.7 quic kly implies co vers most of the edges sets aside edges b efore to absorb leftov er giv es vortex iterativ ely co vers every- thing of a level in the vortex sets aside edges b efore to absorb leftov er of last level implies once degrees are ﬁxed ﬁxes degrees using turns Figure 2: Overall pro of structure 6 optimal by Observ ation 1.2 . This reduction is an essen tial step in our translation of the problem on diameter of simplicial complexes to design theory . The pro of of Theorem 1.8 is inspired by the approach of Glo ck, Jo os, Kühn, and Osthus [ 12 ] for pro ving the conjecture of Ch ung, Diaconis, and Graham [ 4 ] that K ( d ) n has a tigh t d -uniform Euler tour when  n − 1 d − 1  is divisible by d and n is large enough. V ery roughly sp eaking, their approac h works as follows. First, they construct a relatively short closed tight w alk in which ev ery ordered ( d − 1) -set of vertices app ears at least once consecutiv ely . This w alk serves as a “backbone” of the ﬁnal tour. Afterw ards, they remo ve the edges of this tour and decomp ose the remaining edges into tight cycles. This second step is ac hieved b y applying a deep result from design theory [ 13 ], a generalization of Keev ash’s existence of designs [ 18 ] to decomp ositions into arbitrary hypergraphs F . The simple but crucial p oint is then that these cycles can b e merged into the initial tour to form one tour. The analogue of this argumen t do es not work in the extra-tight setting. While one could still ﬁnd a “bac kb one” extra-tight trail and could also apply the F -decomposition theorem to decomp ose into extra- tigh t cycles, the problem is that those extra-tight cycles cannot b e simply merged in to the backbone trail. Indeed, due to the edges that skip a vertex in the sequence, some edges that were cov ered b efore by either the bac kb one trail or the extra-tight cycle will not b e used anymore, while new edges are needed that migh t not b e a v ailable for this. One of the main nov el ingredien ts in our approach is the construction of switchers . These are ﬂexible structures, akin to absorb ers, that allow us to reverse the side eﬀects of the merging op eration. The con- struction of these switchers is int ricate, and again sho wcases the increase in diﬃculty when transitioning from the 2 -dimensional case to higher dimensions: F or d = 2 , one can construct the necessary switc hers explicitly , but for d ≥ 3 , this seems infeasible. A new challenge that arises with this is that we cannot build these switchers for all p ossible lo cations b ecause of space barriers. T o o v ercome this, we will employ the iterative absorption metho d. This will enable us to ensure that the ﬁnal merging step of inserting extra-tight cycles in to a bac kb one trail is only needed on a set U ℓ of constan t size. F or this, in a ﬁrst step, a v ortex is constructed, which is a sequence of nested vertex subsets, the ﬁnal one b eing U ℓ . In each step, the lefto ver is pushed into the next subset. This is achiev ed by the Cov er Down Lemma, which in turn relies crucially on an Appro ximate Decomp osition Lemma similar to the asymptotic solutions in [ 3 , 7 ]. How ever, for the iterative pro cedure to not get out of con trol, we need an Approximate Decomp osition Lemma with “b o osted” error parameters. In spirit, this is similar to the Bo ost Lemma from [ 13 ], and the implemen tation is inspired by the recen t w ork in [ 11 ]. The pro of of Theorem 1.8 stretc hes ov er several sections. In Section 3 , we ﬁrst state the three key lemmas (the Appro ximate Decomp osition Lemma, the Cov er Do wn Lemma, and the Absorb er Lemma) and use them to derive Theorem 1.8 . The pro ofs of these lemmas are then done in the subsequent sections. The Absorb er Lemma is pro ved in Section 4 , the Approximate Decomp osition Lemma in Section 6 , and the Cov er Down Lemma in Section 7 . The latter t wo use the concept of ro oted embeddings from [ 13 ]. In Section 5 , w e quic kly repeat their deﬁnition and state the lemmas that will b e used. 1.3 Notation and Probabilistic T o ols W e will follow the same notation as [ 13 ]. F or completeness, we state them again here. F or a p ositive integer, let [ n ] : = { 1 , . . . , n } and [ n ] 0 : = { 0 , 1 , . . . , n } . F or a set X and i ∈ N 0 , let  X i  b e the set of all i -subsets of X . By the hierarc hy x ≪ y w e mean that for every y ∈ (0 , 1) , there is an x 0 ∈ (0 , 1) suc h that for all 0 < x < x 0 , the subsequent statement holds. If a hierarch y contains sev eral ≪ , they are to be read from righ t to left. F urthermore, if a hierarch y contains 1 /x , x is assumed to b e a p ositive in teger. A d -gr aph H is a d -uniform hypergraph, where w e often iden tify H with its edge set E ( H ) . If G is a d -graph and S ⊆ V ( G ) a set with | S | ∈ [ d ] 0 , the link gr aph G ( S ) is the ( d − | S | ) -graph that has vertex set V ( G ) \ S and contains all subsets S ′ suc h that S ∪ S ′ ∈ G . W e write deg ( S ) : = | G ( S ) | . F or i ∈ [ d − 1] 0 , the minimum resp. maximum i -de gr e e δ i ( G ) resp. ∆ i ( G ) is the minimum resp. maxim um v alue of deg ( S ) o ver all S ∈  V ( G ) i  . Moreov er, δ ( G ) : = δ d − 1 ( G ) and ∆( G ) : = ∆ d − 1 ( G ) . Finally , if S ∈  V ( G ) d − 1  and U ⊆ V ( G ) , deg G ( S, U ) is the num b er of u ∈ U such that S ∪ { u } ∈ G . The hypergraphs of extra-tight trails and extra-tight tours with no rep eated vertex will b e esp ecially imp ortan t in our constructions. Deﬁnition 1.9. The d -graphs S ( d ) T and C ( d ) T created from a sequence T = ( v 1 , . . . , v k ) of pairwise distinct 7 v ertices is called the ( d -uniform) tight p ath , denoted b y P ( d ) k , and the ( d -uniform) tight cycle , denoted b y C ( d ) k , respectively . The shado ws ∂ P ( d +1) k and ∂ C ( d +1) k are called the ( d -uniform) extr a-tight p ath and extr a-tight cycle and are denoted b y E P ( d ) k and E C ( d ) k , resp ectively . W e will need to pay particular atten tion to the ends of paths and extra-tight trails. Deﬁnition 1.10. If T is an extra-tigh t trail, then a set S ⊆ V ( H ) app e ars at an end of T if S ⊆ { v 1 , . . . , v d } or S ⊆ { v k − d +1 , . . . , v k } and analogously for an extra-tight path if T do es not rep eat v ertices. W e will use the following version of a Chernoﬀ b ound sev eral times. Theorem 1.11 ([ 15 ], Corollary 2.3) . L et X : = P i ∈ [ n ] X i wher e the X i ar e indep endently distribute d in [0 , 1] for i ∈ [ n ] . Then, for every 0 < β ≤ 3 / 2 , we have P  | X − E [ X ] | ≥ β E [ X ]  ≤ 2 exp  − β 2 3 E [ X ]  . A very similar result holds for hypergeometric distributions as well: Theorem 1.12 ([ 15 ], Theorem 2.10, Equation (2.6)) . L et X have a hyp er ge ometric distribution 1 with p ar ameters N , n , m . Then for al l 0 ≤ t , we have P  X ≤ E [ X ] − t  ≤ exp  − t 2 2 E [ X ]  . (5) A t one p oint, w e will need the following v arian t whic h is called McDiarmid’s Inequality and can also b e derived from the Azuma-Ho eﬀding Inequality . Theorem 1.13 ([ 15 ], Remark 2.28) . L et X 1 , . . . , X n b e indep endent r andom variables with X i taking values in A i for e ach i ∈ [ n ] . F urthermor e, let c 1 , . . . , c n b e p ositive numb ers and let f : A 1 × · · · × A n → R b e a function satisfying | f ( x ) − f ( ¯ x ) | ≤ c i whenever the ve ctors x, ¯ x ∈ A 1 × · · · × A n diﬀer only in the i -th c o or dinate. Then, the r andom variable Y : = f ( X 1 , . . . , X n ) satisﬁes P  | Y − E [ Y ] | > t  ≤ 2 exp  − 2 t 2 P n i =1 c 2 i  for al l t > 0 . 2 Pro of of Theore m 1.1 and 1.7 In this section, we show how the main Theorem 1.1 and Theorem 1.7 can b e derived from Theorem 1.8 . The remainder of the pap er will then concern the pro of of Theorem 1.8 . First, we need an observ ation that justiﬁes the degree requirement ( 4 ). Observ ation 2.1. If C = ( v 1 , . . . , v k ) is an extr a-tight tour in a d -gr aph with k ≥ d + 3 , then deg C ( v ) ≡ 0 (mo d d 2 ) for al l v ∈ V ( C ) . If, mor e over, C is an extr a-tight cycle, then deg C ( v ) = d 2 . If T = ( v 1 , . . . , v k ) is an extr a-tight tr ail in a d -gr aph with k ≥ 2 d wher e the 2 d vertic es at the ends of T ar e p airwise distinct, then deg T ( v ) ≡ ( i ( d − 1) + 1 ∃ i ∈ [ d ] : v ∈ { v i , v k +1 − i } 0 else (mo d d 2 ) . If, mor e over, T is an extr a-tight p ath, then deg T ( v ) = ( i ( d − 1) + 1 ∃ i ∈ [ d ] : v ∈ { v i , v k +1 − i } d 2 else. 1 Recall that a random v ariable X has hypergeometric distribution with parameters N , n , m if X : = | S ∩ [ m ] | where S is a subset chosen uniformly at random among all subsets of [ N ] of size n . 8 Pr o of. Given an edge e in C , k ≥ d + 3 implies that at least 3 vertices are not in e . By the deﬁnition of extra-tigh tness, each edge can skip at most one v ertex. Therefore, all v ertices cov ered b y e are consecutiv e v ertices on C except for at most one. Thus, there are unique ι ∈ [ k ] , σ ∈ [ d ] such that e = e ι,σ (cf. Deﬁnition 1.4 ). Hence, k ≥ d + 3 implies that the e ι,σ are pairwise distinct. The lab el v ℓ is exactly in the e ι,σ with ι ∈ { ℓ − d, ℓ − d + 1 , . . . , ℓ } and σ ∈ [ d ] \{ ℓ − ι } (where e ι,σ : = e ι + k,σ for − d < ι < 0 ). These are exactly ( d + 1)( d − 1) + 1 = d 2 edges (the +1 comes from ι = ℓ where σ ∈ [ d ] ). Thus, deg C ( v ) ≡ 0 (mo d d 2 ) for all v ∈ V ( C ) . Next, we consider the extra-tigh t trail T . If v ℓ is a lab el that is not at the end of T , then v ℓ is in the same d 2 edges as abov e. Since the extra-tight trails are symmetric, we only ha ve to consider v ℓ with ℓ ∈ [ d ] . There, the lab el v ℓ is exactly in the e ι,σ with ι ∈ [ ℓ ] and σ ∈ [ d ] \{ ℓ − ι } (since k ≥ 2 d , [ ℓ ] ⊆ [ k − d ] whence all these e ι,σ are indeed deﬁned). Therefore, v ℓ is in ℓ ( d − 1) + 1 edges. Since the vertices at the end are pairwise distinct, the result follo ws. F requently , we will ha ve to connect several extra-tight trails into one long extra-tigh t trail. The follo wing lemma allows us to do this. Lemma 2.2. L et n , d , k , ℓ b e non-ne gative inte gers with d ≥ 2 . L et A = ( a 1 , . . . , a k ) and B = ( b 1 , . . . , b ℓ ) b e the (p ossibly empty) se quenc es of two extr a-tight tr ails in an n -vertex gr aph G such that A ∩ B = ∅ . F urthermor e, assume that U ⊆ V ( G ) such that | U | > d 2  n − δ ( G ) + ∆( A ) + ∆( B ) + 2 d 3  + 4 d. Then ther e ar e 2 d distinct vertic es v 1 , . . . , v 2 d ∈ U such that ( a 1 , . . . , a k , v 1 , . . . , v 2 d , b 1 , . . . , b ℓ ) is the se quenc e of an extr a-tight tr ail in G . Pr o of. W e select the vertices v 1 , . . . , v 2 d one after the other. F or i = 1 , . . . , 2 d , let E i b e the link of v i in the subgraph of the extra-tight trail ( a 1 , . . . , a k , v 1 , . . . , v 2 d , b 1 , . . . , b ℓ ) induced b y the vertex set { a 1 , . . . , a k , v 1 , . . . , v i , b 1 , . . . , b ℓ } . Note that, by Observ ation 2.1 , | E i | ≤ d 2 . No w supp ose, for i ∈ [2 d − 1] , we ha ve found vertices v 1 , . . . , v i − 1 ∈ U \{ a k − d +1 , . . . , a k , b 1 , . . . , b d } suc h that A i := ( a 1 , . . . , a k , v 1 , . . . , v i − 1 ) is an extra-tight trail that is disjoint from B . This clearly holds for i = 0 , where A 0 = A . W e pick v i ∈ U \{ a k − d +1 , . . . , a k , v 1 , . . . , v i − 1 , b 1 , . . . , b d } such that { v i } ∪ e is an edge of G i := G \ ( A i ∪ B ) for each e ∈ E i . With δ ( G i ) ≥ δ ( G ) − ∆( A i ) − ∆( B ) ≥ δ ( G ) − ∆( A ) − ∆( B ) − 2 d · d 2 this is p ossible, b ecause      \ S ∈ E i G i ( S ) ! \{ a k − d +1 , . . . , a k , v 1 , . . . , v i − 1 , b 1 , . . . , b d }      ≥      \ S ∈ E i G i ( S )      − 4 d ≥ n − | E i | ·  n − δ ( G ′ )  − 4 d ≥ n − d 2 ·  n − δ ( G ) + ∆( A ) + ∆( B ) + 2 d 3  − 4 d > n − | U | . F or i ≤ 2 d − 2 , as all new edges are contained in { a k − d +1 , . . . , a k , v 1 , . . . , v i − 1 } and these v ertices are all diﬀeren t, it follows that A i +1 := ( a 1 , . . . , a k , v 1 , . . . , v i ) is an extra-tight trail that is disjoint from B and w e can contin ue to the next i . F or i = 2 d − 1 , b esides the edges of the extra-tight trail ( a 1 , . . . , a k , v 1 , . . . , v 2 d − 1 ) , we no w also identiﬁed in G 2 d − 1 the edges { v d − i , . . . , v 2 d − 1 , b 1 , . . . , b d − i } for i = 1 , . . . , d − 1 , which are pairwise diﬀerent, b ecause they are con tained in { v d +1 , . . . , v 2 d − 1 , b 1 , . . . , b d } and these are all diﬀerent. It only remains to pic k v 2 d ∈ U \{ a k − d +1 , . . . , a k , v 1 , . . . , v 2 d − 1 , b 1 , . . . , b d } such that { v 2 d } ∪ e is an edge of G 2 d := G \ ( A 2 d ∪ B ) for each e ∈ E 2 d . This is p ossible by the same calculation. Moreov er, all the edges { v 2 d } ∪ e for e ∈ E 2 d are pairwise diﬀeren t b ecause they are contained in { v d +1 , . . . , v 2 d , b 1 , . . . , b d } and these are all diﬀeren t. Hence we indeed get an extra-tight trail ( a 1 , . . . , a k , v 1 , . . . , v 2 d , b 1 , . . . , b ℓ ) . W e start by proving Theorem 1.7 b efore proving Theorem 1.1 at the end of this section. 9 Pr o of of The or em 1.7 . Let G b e an n -v ertex d -graph with δ ( G ) ≥ (1 − α ) n where every vertex degree is divisible by d 2 . First, we wan t to ﬁnd an extra-tigh t path P = ( v 1 , . . . , v 2 d ) in G . W e can do this by applying Lemma 2.2 with A 2 . 2 : = B 2 . 2 : = () and U 2 . 2 = V ( G ) . No w consider the graph G ′ : =  G \ P  ∪  { v 1 , . . . , v d } , { v d +1 , . . . , v 2 d }  . Since we assumed that deg G ( v ) ≡ 0 (mo d d 2 ) holds for all v ∈ V ( G ) and by Observ ation 2.1 , we get, for i ∈ [ d ] deg G ′ ( v i ) ≡ −  i ( d − 1) + 1  + 1 = − i ( d − 1) ≡ ( d + 1 − i )( d − 1) + 1 (mo d d 2 ) and similarly , for i ∈ { d + 1 , . . . , 2 d } . W e can conclude that G ′ satisﬁes ( 4 ) with the ends ( v d , v d − 1 , . . . , v 1 ) and ( v 2 d , . . . , v d +1 ) . F urthermore, δ ( G ′ ) ≥ (1 − 2 α ) n if n is large enough. Thus, we can apply Theorem 1.8 with α 1 . 8 : = 2 α and get an extra-tight trail T = ( v d , . . . , v 1 , . . . , v 2 d , . . . , v d +1 ) with ends ( v d , . . . , v 1 ) and ( v 2 d , . . . , v d +1 ) cov ering all edges of G ′ . But together with the edges of P , the vertex sequence of T forms an extra-tight tour that cov ers all edges of G . Next, w e will pro ve Theorem 1.1 . F or this, we will ﬁnd a simplicial d -complex C on [ n ] whose dual graph G ( C ) is a path and where all but at most d − 1 of the sets in  [ n ] d  app ear in C . The simplest wa y to do this w ould b e to deﬁne a straight simplicial d -complex with the required prop erties. Ho wev er, this is not alw ays p ossible: by Observ ation 2.1 , in a straight simplicial d -complex C , the n umber of d -sets a ﬁxed elemen t i ∈ [ n ] app ears in is divisible by d 2 unless i is among the 2 d elemen ts that o ccur at one of the ends of C . Therefore, if the vertex degree  n − 1 d − 1  in K ( d ) n is not divisible b y d 2 , then it is imp ossible that a straight sim plicial complex has the maxim um p ossible diameter. F ortunately , we can adjust this degree issue by ﬁrst deﬁning a relativ ely short simplicial d -complex C 1 that uses “turns”. Afterw ards, the graph of the d -sets that are not elements in C 1 (min us at most d − 1 edges) will satisfy ( 4 ) suc h that w e can co ver it with a straight simplicial d -complex C 2 . W e will make sure that C 1 and C 2 ha ve the same ends so that they can b e combined into one long simplicial d -complex of maximum diameter. T o get C 1 , we will start with a straight simplicial d -complex and insert turns one after the other. The follo wing observ ation speciﬁes how we insert turns and ho w this inﬂuences the num b er of d -sets of the simplicial complex a ﬁxed elemen t is contained in. v 1 v 2 v 3 v d +2 v d +3 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v d +2 v d +3 v 7 v 8 v 9 v 10 Figure 3: The simplicial d -complex C ′ for d = 2 and d = 3 . Observ ation 2.3. L et C b e a simplicial d -c omplex on [ n ] with fac et set F whose dual gr aph G ( C ) is a p ath P . L et d + 4 c onse cutive vertic es of P b e the fac ets of the str aight simplicial d -c omplex ( v 1 , . . . , v 2 d +4 ) . F urthermor e, assume that C do es not c ontain any of the d -sets { v d +2 , v d +4 , v d +5 , . . . , v 2 d +3 }\{ v i } with i ∈ { d + 4 , . . . , 2 d + 2 } . L et C ′ b e the simplicial d -c omplex gener ate d by F ′ : = F \  { v d +3 , . . . , v 2 d +3 }  ∪  { v d +2 , v d +4 , . . . , v 2 d +3 }  . Then G ( C ′ ) is also a p ath and | ( ∂ F )( v ) | − | ( ∂ F ′ )( v ) | =      − ( d − 1) v = v d +2 d − 1 v = v d +3 0 else. Figure 3 sho ws C ′ in the case d = 2 and d = 3 where C is just the straigh t simplicial d -complex ( v 1 , . . . , v 2 d +4 ) . Notice that, in the left ﬁgure, the v ertex v d +2 is con tained in 5 = 4 + ( d − 1) edges, whereas 10 v d +3 is con tained in 3 = 4 − ( d − 1) . Similarly , in the righ t ﬁgure, v d +2 is con tained in 11 = 9 + ( d − 1) triangles and v d +3 is contained in 7 = 9 − ( d − 1) triangles. Pr o of of Observation 2.3 . Since C do es not con tain an y of the d -sets { v d +2 , v d +4 , v d +5 , . . . , v 2 d +3 }\{ v i } with i ∈ { d + 4 , . . . , 2 d + 2 } , the only facets in C ′ that hav e an intersection of size d − 1 with the facet { v d +2 , v d +4 , v d +5 , . . . , v 2 d +3 } are { v d +2 , . . . , v 2 d +2 } and { v d +4 , . . . , v 2 d +4 } , i.e. the same facets that had an intersection of size ( d − 1) with { v d +3 , . . . , v 2 d +3 } . Therefore, G ( C ′ ) is still a path. The d -sets that C con tains but not C ′ are { v d +3 , v d +4 , v d +5 , . . . , v 2 d +3 }\{ v i } with i ∈ { d + 4 , . . . , 2 d + 2 } . The d -sets that C ′ con tains but not C are { v d +2 , v d +4 , v d +5 , . . . , v 2 d +3 }\{ v i } with i ∈ { d + 4 , . . . , 2 d +2 } . Pr o of of The or em 1.1 . Let G b e the complete d -graph on the vertex set [ n ] . W e will successively build a simplicial d -complex C on [ n ] . If a d -set { i 1 , . . . , i d } appears in C , we will delete it from the edge set of G . That wa y , G will alwa ys encode the set of d -sets that still need to app ear in C . Our ﬁnal simplicial d -complex will mostly b e a straigh t simplicial d -complex but with a few turns throughout. It will start with a straight simplicial d -complex starting with (1 , . . . , d ) and end with a straigh t simplicial d -complex ending with ( n − d + 1 , . . . , n ) . T o b e able to do that, we m ust make sure that the d -sets { 1 , . . . , d } and { n − d + 1 , . . . , n } are not used anywhere else. Note that the d -uniform Handshak e Lemma implies that the sum of vertex degrees n  n − 1 d − 1  in G is divisible by d . Let s ∈ [ d − 1] 0 b e such that n  n − 1 d − 1  ≡ d ( s + 1) (mo d d 2 ) . Let M be a set of s disjoin t edges of G that do not contain an y of 1 , . . . , d, n − d + 1 , . . . , n . Delete M from G . Now, the sum of 1-degrees in G is ≡ d (mo d d 2 ) . The edges in M will b e the only d -sets not co vered by our ﬁnal simplicial d -complex. This together with Observ ation 1.2 (i) then the simplicial d -complex has the diameter  1 d  n d  − d +1 d  as desired. Next, we will adjust the vertex degrees in G using turns. By Observ ation 2.3 , each turn “mov es” d − 1 from one v ertex degree to another vertex degree. Our goal no w is to ﬁnd a set of pairs of vertices that enco de how one could ﬁx the degree with these turns. Claim. There is a digraph D on the vertex set V ( G ) such that the follo wing conditions hold: (i) for all v ∈ V ( G ) , if deg + D ( v ) is the n umber of outgoing edges of v , and deg − D ( v ) the num b er of incoming edges of v , then deg G ( v ) +  deg + D ( v ) − deg − D ( v )  ( d − 1) ≡ ( i ( d − 1) + 1 ∃ i ∈ [ d ] : v ∈ { i, n + 1 − i } 0 else (mo d d 2 ); (ii) for all v ∈ V ( G ) , deg + D ( v ) + deg − D ( v ) ≤ 5 d 2 ; (iii) if d = 2 , then for all ( u, v ) ∈ D , { u, v } ∈  M ∪  { 1 , 2 } , { n − 1 , n }  . Pr o of of claim: Start with D = ∅ . Then (ii) and (iii) are already satisﬁed. W e will go from v = 1 to v = n and satisfy (i) for v while maintaining the other conditions and (i) for all u with u < v adding at most 2 d 2 edges to D . Thus, throughout, D will hav e at most 2 d 2 n edges. Supp ose, w e wan t to satisfy (i) for v next where v ∈ [ n − 1] . Let a ∈ [ d 2 − 1] 0 b e the n umber suc h that deg G ( v ) + ( a + deg + ( v ) − deg − ( v ))( d − 1) ≡ ( i ( d − 1) + 1 ∃ i ∈ [ d ] : i ∈ { i, n + 1 − i } 0 else. (mo d d 2 ) Note that this n umber exists b ecause d − 1 and d 2 are coprime. W e wan t to add a more edges to D where v is the ﬁrst element. F or this, let T be the set of vertices that already app ear in 3 d 2 − 1 edges of D or already appear in an edge with v or v + 1 in D . Since | D | ≤ 2 d 2 n , T will contain at most 2 2 . 9 n + 2 · 5 d 3 v ertices. Next, pick a distinct vertices u 1 , . . . , u a in V ( G ) \  T ∪ { v , v + 1 , 1 , 2 , n − 1 , n } ∪ S e ∈ M e  and add the 2 a ≤ 2 d 2 edges ( v , u j ) , ( u j , v + 1) with j ∈ [ a ] to D . One can quickly see that (iii) and (ii) are all still satisﬁed, and (i) is no w satisﬁed for 1 , . . . , v . Con tinuing that w ay , we can satisfy (i) for all 1 , . . . , n − 1 . W e claim that then (i) is also satisﬁed for n . Indeed, using that P v ∈ V ( G ) deg G ( v ) ≡ d (mo d d 2 ) and P v ∈ V ( G ) (deg + ( v ) − deg − ( v ))( d − 1) = 0 , 11 w e get d ≡ X v ∈ V ( G )  deg G ( v ) + (deg + D ( v ) − deg − D ( v ))( d − 1)  (i) = deg G ( n ) +  deg + D ( n ) − deg − D ( n )  ( d − 1) + 2 d X i =1  i ( d − 1) + 1  − d = deg G ( n ) +  deg + D ( n ) − deg − D ( n )  ( d − 1) + d ( d + 1)( d − 1) + 2 d − d ≡ deg G ( n ) +  deg + D ( n ) − deg − D ( n )  ( d − 1) . This concludes the pro of of the claim. − Let D be the digraph given b y the claim and let D = { p 1 , . . . , p t } . W e will no w start to build the simplicial complex b y deﬁning a sequence Seq =  v 0 d +5 , v 0 d +6 . . . , v 0 2 d +4 , v 1 1 , v 1 2 , . . . , v 1 2 d +4 , v 2 1 , . . . , v 2 2 d +4 , . . . , v t 1 , . . . , v t 2 d +4  suc h that the follo wing conditions are satisﬁed: 1) v 0 d +5 = 1 , v 0 d +6 = 2 , . . . , v 0 2 d +4 = d ; 2)  v t d +5 , . . . , v t 2 d +4  ∩ { n + 1 − d, . . . , n } = ∅ ; 3) if p i = ( j 1 , j 2 ) , then v i d +2 = j 1 and v i d +3 = j 2 for all i ∈ [ t ] ; 4) all the d -sets of the straigh t simplicial d -complex deﬁned via the sequence Seq and all the d -sets of the form { v i d +2 , v i d +4 , v i d +5 , . . . , v i 2 d +3 }\{ v i j } with j ∈ { d + 4 , . . . , 2 d + 2 } are pairwise distinct and in G \  { n − d + 1 , . . . , n }  ; 5) no element of [ n ] app ears in Seq more than 24 d 3 times. W e start by deﬁning v 0 d +5 , v 0 d +6 . . . , v 0 2 d +4 , v 1 d +2 , v 1 d +3 , v 2 d +2 , v 2 d +3 , . . . , v t d +2 , v t d +3 suc h that 1) and 3) are satisﬁed. Note that this will not violate 4) since w e ensured that { 1 , 2 , . . . , d } is still an edge in G . F or d = 2 , we hav e to b e more careful here, but prop erty (iii) of the claim ensures that every 2-set of 4) that is already fully deﬁned is unique and in G \  { n − 1 , n }  . The remaining elemen ts of Seq are deﬁned from left to right. Suppose w e w ant to deﬁne the element v i j next with i ∈ [ t ] and j ∈ [2 d + 4] . Let V ′ b e the set of elements in [ n ] that already app ear 24 d 3 man y times in Seq . Since Seq consists of d + t · (2 d + 4) ≤ 12 d 3 n elements, V ′ con tains at most 12 d 3 n 24 d 3 = n 2 elemen ts. Let E ′ b e the set of d -sets mentioned in 4) . Let E ′′ ⊆ E ′ b e the subset of d -sets that are already completely deﬁned. F or each e ∈ E ′ that contains v i j where the other d − 1 elemen ts of e are already deﬁned, let V e b e the set of elements v ∈ [ n ] where if we deﬁned v i j to be v , e would b ecome an edge that is already in E ′′ or not in G . There can b e at most ( d 2 + d ) such e and each V e has a size of at most 24 d 3 · ( d 2 + d ) ≤ 25 d 5 . Deﬁne v i j to b e an y elemen t of [ n ] \  V ′ ∪ S v i j ∈ e ∈ E ′ V e ∪ { n + 1 − d, . . . , n }  that is not also already an elemen t of Seq that app ears among the d elements before or after v i j . If n is large enough with resp ect to d , we alwa ys hav e an option for v i j . Thus, we can deﬁne the sequence Seq suc h that it fulﬁlls the required prop erties. Let C b e the straigh t simplicial d -complex with v ertex sequence Seq . F or eac h i ∈ [ t ] w e apply Observ ation 2.3 on C and the v ertex sequence v i 1 , . . . , v i 2 d +4 . By 4) , the required d -sets are missing from C . W e end up with a simplicial d -complex C 1 . By 4) again, all d -sets of C 1 are in G . Let G ′ b e the d -graph that contains all edges of G that are not d -sets in C 1 . F urthermore, w e add the edge  v t d +5 , . . . , v t 2 d +4  to G ′ . By Observ ation 2.3 , 2) , 3) , and prop ert y (i) of the claim, w e no w ha ve deg G ′ ( v ) ≡ ( i ( d − 1) + 1 ∃ i ∈ [ d ] : v ∈  v t d +4+ i , n + 1 − d  0 else (mo d d 2 ) . F urthermore, G ′′ has minimum ( d − 1) -degree at least n − 24 d 3 · ( d 2 + d − 1) b y 5) . Thus, we can apply Theorem 1.8 and get an extra-tight trail with ends  v t d +5 , v t d +6 , . . . , v t 2 d +4  and ( n − d + 1 , . . . , n ) . But 12 this corresp onds to a straigh t simplicial d -complex C 2 whic h we can combine with C 1 to get a simplicial d -complex whose dual graph is a path and which has every element of  [ n ] d  \ M as d -sets. Since these are all but at most d − 1 many d -sets, Observ ation 1.2 shows that the simplicial d -complex has diameter  1 d  n d  − d + 1 d  . 3 Pro of of Theore m 1.8 In this section, w e will ﬁrst state all the key lemmas, i.e. the Appro ximate Decomp osition Lemma, the Co ver Do wn Lemma, and the Absorb er Lemma. Afterwards, w e will prov e Theorem 1.8 assuming all the stated lemmas. The pro ofs of the three main lemmas will follo w in subsequent sections. 3.1 Key Lemmas The result of Bohman and Newman [ 3 ] already yields (in the complete d -graph) an extra-tight trail that co vers all but a negligible prop ortion of the edges. Our ﬁrst key lemma is a strengthening of this. Namely , w e need to ﬁnd suc h an extra-tight trail in d -graphs with minim um degree δ ( G ) ≥ (1 − α ) n . While it ma y app ear plausible that for v ery small α , the analysis of Bohman and Newman still works, this w ould eﬀect the size of the leftov er. The crucial feature of our result is that w e can ensure the lefto ver to b e still arbitrarily small, that is, the qualit y of the appro ximate decomposition do es not depe nd on α . Lemma 3.1 (Approximate Decomp osition Lemma) . Supp ose 1 /n ≪ γ ≪ α ≪ 1 /d ≤ 1 / 2 . L et G b e an n -vertex d -gr aph with δ ( G ) ≥ (1 − α ) n . Then ther e exists an extr a-tight tr ail such that the leftover L of al l e dges of G not c over e d by the tr ail satisﬁes ∆( L ) ≤ γ n . Here, the maximum degree condition on L ensures that the n umber of uncov ered edges is less than γ n d , and that these edges are not concentrated at a few vertices. With this lemma, we end up with one extra-tight trail that cov ers most of the edges of G , but there we are stuck, unless we prepared well ahead some extra uncov ered structure that would help us to contin ue. This is exactly the idea of the iterative absorption method. F or that, one builds randomly a “vortex”, i.e. a sequence V ( G ) = U 0 ⊇ U 1 ⊇ · · · ⊇ U ℓ of subsets with natural regularity prop erties, suc h that | U i | / | U i +1 | and | U ℓ | are constan t. In eac h iteration i of the extra-tight trail-building pro cess, we use the Approximate Decomp osi- tion Lemma and additional arguments to ﬁnds an extra-tight trail co vering all unco vered edges in G [ U i ] \ G [ U i +1 ] while not using to o many edges of G [ U i +1 ] . Here it is crucial that we can bo ost the parameters, meaning the maxim um degree of the extra-tight trail in G [ U i +1 ] is m uch smaller than the maxim um degree of the complemen t of G [ U i +1 ] . This will ensure that at the end of this pro cess, al l edges in G \ G [ U ℓ ] are co vered, while almost all edges in G [ U ℓ ] are unco vered. The follo wing Cov er Down Lemma captures the iterative step. There w e hav e to assume that certain degree conditions are fulﬁlled to b e able to cov er all desired edges by an extra-tigh t trail. Later, when we apply the Cov er Do wn Lemma in the pro of of Theorem 1.8 , these degree conditions will b e automatically satisﬁed by the fact that we started with a graph that satisﬁes ( 4 ). Lemma 3.2 (Cov er Down Lemma) . Supp ose 1 /n ≪ α, µ ≪ 1 /d ≤ 1 / 2 with α < µ 1 . 1 . L et G b e a d -gr aph on n vertic es and U ⊆ V ( G ) with | U | = µn . Supp ose that δ ( G ) ≥ (1 − α ) n and deg ( v ) ≡ 0 (mo d d 2 ) holds for al l v ∈ V ( G ) \ U . Then ther e exists an extr a-tight tr ail T whose ends ar e in U and such that G \ G [ U ] ⊆ T ⊆ G and ∆( T [ U ]) ≤ µ 2 n . T o also incorp orate the lefto ver in U ℓ in to our extra-tight trail, we wan t to use absorb ers. In our case, this means we set aside an extra-tight trail b efore using the Co ver Do wn Lemma that can absorb the lefto ver edges. Ho wev er, we do not kno w b eforehand how the leftov er will lo ok like. Hence, the absorb er m ust work for all p ossible leftov ers. There, U ℓ b eing of constan t size is vital b ecause this allo ws us to build an absorb er that w orks for all possibilities. Lemma 3.3 (Absorb er Lemma) . L et 1 /n ≪ η ≪ 1 /m ≪ α ≪ 1 /d ≤ 1 / 2 . L et G b e a d -gr aph on n vertic es with δ ( G ) ≥ (1 − α ) n and U ⊆ V ( G ) a vertex set of size m . Then G c ontains an extr a-tight tr ail A such that 13 • V ( A ) ⊇ U and U is an indep endent set in A ; • ∆( A ) ≤ ηn ; • for e ach d -gr aph L ⊆ G [ U ] with δ ( L ) ≥ (1 − α ) m wher e e ach 1-de gr e e is divisible by d 2 , ther e is an extr a-tight tr ail with e dge set L ∪ A having the same ends as A . With some extra work, one could omit the minimum degree requirement on L . How ev er, since w e get this for free in our main proof, we omit this here. 3.2 Pro of of Theorem 1.8 In this subsection, w e will prov e our main Theorem 1.8 assuming the Absorb er Lemma and the Cov er Do wn Lemma. While the Approximate Decomp osition Lemma is not needed in this pro of, w e will need it when w e pro ve the Cov er Do wn Lemma. W e start by formally deﬁning the concept of a vortex and proving its existence in G . Deﬁnition 3.4. Let G be a d -graph on n vertices. An ( α, µ, m ) -vortex in G is a sequence U 0 ⊇ U 1 ⊇ · · · ⊇ U ℓ suc h that (i) U 0 = V ( G ) ; (ii) | U i | = ⌊ µ | U i − 1 |⌋ for all i ∈ [ ℓ ] ; (iii) | U ℓ | = m ; (iv) deg G ( S, U i ) ≥ (1 − α ) | U i | for all i ∈ [ ℓ ] and S ∈  U i − 1 d − 1  . The following lemma and proof are similar to Lemma 3.7 in [ 1 ] where this is done for graphs instead of hypergraphs. Lemma 3.5. L et 1 /m ′ ≪ α, µ ≪ 1 /d ≤ 1 / 2 . Supp ose G is a d -gr aph on n ≥ m ′ vertic es with δ ( G ) ≥ (1 − α ) n . Then G has an ( α + µ 2 , µ, m ) -vortex for some ⌊ µm ′ ⌋ ≤ m ≤ m ′ . Pr o of. W e deﬁne n 0 : = n and n i : = ⌊ µn i − 1 ⌋ . Note µ i n ≥ n i ≥ µ i n − 1 1 − µ. . Let ℓ : = 1 + max { i ≥ 0 : n i ≥ m ′ } and let m : = n ℓ . By deﬁnition, ⌊ µm ′ ⌋ ≤ m ≤ m ′ . Finally , for i ∈ [ ℓ ] , let µ i : = n − 1 / 3 i X j =1 µ − ( j − 1) / 3 = n − 1 / 3 µ − i/ 3 − 1 µ − 1 / 3 − 1 ≤  µ i − 1 n  − 1 / 3 1 − µ 1 / 3 ≤ m ′− 1 / 3 1 − µ 1 / 3 ≤ µ 2 3 , (6) and let µ 0 : = 0 . Supp ose that for some i ∈ [ ℓ ] , w e hav e already found an ( α + 3 µ i − 1 , µ, n i − 1 ) -v ortex U 0 , . . . , U i − 1 in G . This is true for i = 1 . In particular, δ ( U i − 1 ) ≥ (1 − α − 3 µ i − 1 ) n i − 1 . Let U i b e a subset chosen uniformly at random among all subsets of U i − 1 of size n i . F or each S ∈  U i − 1 d − 1  , we hav e E [deg G ( S, U i )] ∈ [(1 − α − 3 µ i − 1 ) n i , n i ] . Therefore, the hypergeometric v ariant ( 5 ) of the Chernoﬀ b ound implies P h deg G ( S, U i ) ≤  1 − α − 3 µ i − 1 − 2 n − 1 / 3 i − 1  n i i ≤ P h deg G ( S, U i ) ≤ E [deg G ( S, U i )] − 2 n − 1 / 3 i − 1 n i i ≤ exp    −  2 n − 1 / 3 i − 1 n i  2 2 E [deg G ( S, U i )]    ≤ exp    −  2 n − 1 / 3 i − 1 n i  2 2 n i    = exp  − 2 n − 2 / 3 i − 1 n i  ≤ exp  − 2 n − 2 / 3 i − 1 ( µn i − 1 − 1)  ≤ exp  − µn 1 / 3 i − 1  . Th us, if m ′ ≤ n i − 1 is large enough, we can use a union bound o ver all S ∈  U i − 1 d − 1  and get that, with high probability , there is a choice for U i suc h that deg G ( S, U i ) ≥  1 − α − 3 µ i − 1 − 2 n − 1 / 3 i − 1  n i holds for all S ∈  U i − 1 d − 1  . Fix suc h a choice of U i . Then 3 µ i − 1 + 2 n − 1 / 3 i − 1 ≤ 3 n − 1 / 3 i − 1 X j =1 µ − ( j − 1) / 3 + 2 µ − ( i − 1) / 3 n − 1 / 3 ≤ 3 n − 1 / 3   i − 1 X j =1 µ − ( j − 1) / 3 + µ − ( i − 1) / 3   = 3 µ i 14 implies that U 0 , . . . , U i form a ( α + 3 µ i , µ, n i ) -v ortex in G . Rep eating this for all i ∈ [ ℓ ] , we ﬁnally obtain an ( α + 3 µ ℓ , µ, m ) -vortex and because of ( 6 ), µ ℓ ≤ µ 2 / 3 , the lemma follows. W e are no w ready to prov e the main theorem on extra-tigh t Euler trails. The pro of consists of the follo wing steps: Step 1 Fix a v ortex: Apply Lemma 3.5 to get a v ortex U 0 ⊇ U 1 ⊇ · · · ⊇ U ℓ . Step 2 Build an absorber for the ﬁnal set: Apply Lemma 3.3 to obtain an absorb er for U ℓ and connect it to one of the given ends ( v ′ d , . . . , v ′ 1 ) . The other end of the absorb er is extended into U ℓ suc h that it can b e connected at the v ery end when all but the edges in U ℓ are used. W e call the resulting extra-tigh t trail T end . In the following, w e wan t to cov er the edges of G ′ : = G − T end . The other end ( v 1 , . . . , v d ) is extended by d arbitrary vertices to form the extra-tigh t trail T 0 . Step 3 Iterativ e usage of Cov er Down: Giv en an extra-tight trail T i co vering G ′ \ G ′ [ U i ] , w e sho w how this can b e extended to an extra-tigh t trail T i +1 co vering G ′ \ G ′ [ U i +1 ] using the Cov er Do wn Lemma 3.2 . This is illustrated in Figure 4 . Step 4 Final connecting and absorbing the lefto ver. In the end, we hav e an extra-tight trail T ℓ whic h co vers all edges outside of G ′ [ U ℓ ] . W e connect it to the end of T end . By the prop erty of the absorb er, w e can absorb the remaining uncov ered edges into T end , yielding an extra-tigh t trail co vering all edges of G . U 0 \ U i U i \ U i +1 U i +1 \ U i +2 U i +2 u ( i +1) 2 d +1 , . . . , u ( i +1) 3 d u ( i +1) 2 d , . . . , u ( i +1) d +1 u ( i +1) 1 , . . . , u ( i +1) d g ( i +1) 1 , . . . , g ( i +1) 2 d u ( i ) 2 d +1 , . . . , u ( i ) 3 d v d , . . . , v 1 Figure 4: The inductiv e step using the Co ver Down Lemma. By the inductiv e hypothesis, there is a (blue) extra-tigh t trail cov ering G ′ \ G ′ [ U i ] . One end is extended b y a (green) extra-tight trail in to U i +1 \ U i +2 . Then, the Co ver Down Lemma 3.2 gives us an (orange) extra-tight trail cov ering all remaining edges in G ′ [ U i ] \ G ′ [ U i +1 ] . Finally , these tw o trails are connected via a short (red) extra-tigh t trail. Pr o of of The or em 1.8 . Let 1 /n ≪ η ≪ 1 /m ′ ≪ α, µ ≪ 1 /d with µ 1 . 5 < α < 1 2 µ 1 . 1 . Step 1. Apply Lemma 3.5 on G to get an ( α + µ 2 , µ, m ) -vortex U 0 ⊇ U 1 ⊇ · · · ⊇ U ℓ for some ⌊ µm ′ ⌋ ≤ m ≤ m ′ . In particular, we hav e 1 /n ≪ η ≪ 1 /m ≪ α, µ ≪ 1 /d . Step 2. W e apply the Absorb er Lemma 3.3 with the parameters U 3 . 3 : = U ℓ , α 3 . 3 : = 2( µ + α ) , and G 3 . 3 : = G [( U 0 \ U 1 ) ∪ U ℓ ] −  { v 1 , . . . , v d } , { v ′ 1 , . . . , v ′ d }  to get an extra-tigh t trail A = ( a 1 , . . . , a q ) with ends ( a 1 , . . . , a d ) and ( a q − ( d − 1) , . . . , a q ) such that U ℓ is indep endent in A and ∆( A ) ≤ η n . F urthermore, A has the following absorbing prop ert y: F or eac h d -graph L on U ℓ with δ ( L ) ≥ (1 − 4 µ ) m where each 1- degree is divisible b y d 2 , there is an extra-tigh t trail with edge set L ∪ A ha ving the same ends ( a 1 , . . . , a d ) and ( a q − ( d − 1) , . . . , a q ) . Note that since U ℓ is indep endent in A and by the wa y we c hose G 3 . 3 , w e are guaran teed that neither { v 1 , . . . , v d } nor { v ′ 1 , . . . , v ′ d } is in A . Let { g ′ 1 , . . . , g ′ d } ∈ G [ U ℓ ] −  { v 1 , . . . , v d } , { v ′ 1 , . . . , v ′ d }  . W e start building our ﬁnal extra-tigh t trail b y gluing one end of A to one of the giv en ends ( v ′ d , . . . , v ′ 1 ) and the other end of A to some vertices to ( g ′ 1 , . . . , g ′ d ) : by Lemma 2.2 , we can ﬁnd vertices g 1 , . . . , g 4 d ∈ U 0 \ U 1 suc h that  g ′ 1 , . . . , g ′ d , g 1 , . . . , g 2 d , a 1 , . . . , a q , g 2 d +1 , . . . , g 4 d , v ′ d , . . . , v ′ 1  15 is an extra-tight trail T end in G . Note that ∆( T end ) ≤ 2 η n and ∆( T end [ U i ]) ≤ 2 for all i > 0 . Let G ′ : = G \ T end . Step 3. Using Lemma 2.2 , w e greedily pick some u (0) 2 d +1 , . . . , u (0) 3 d ∈ U 0 \ U 1 suc h that  v 1 , . . . , v d , u (0) 2 d +1 , . . . , u (0) 3 d  forms an extra-tight trail T 0 in G ′ . Inductively , assume that for some i ∈ [ ℓ − 1] 0 , we hav e already found an extra-tight trail T i in G ′ with ends ( v 1 , . . . , v d ) and  u ( i ) 2 d +1 , . . . , u ( i ) 3 d  suc h that (i) u ( i ) 2 d +1 , . . . , u ( i ) 3 d ∈ U i (ii) G ′ \ G ′ [ U i ] ⊆ T i ⊆  G ′ \ G ′ [ U i +1 ]  ∪  { v 1 , . . . , v d }  (iii) ∆( T i [ U i ]) ≤ 2 µ 2 | U i | . Note that these conditions are satisﬁed for i = 0 . W e show no w ho w to get the same result for i + 1 instead of i (where U ℓ +1 : = ∅ ). Using Lemma 2.2 , we can ﬁnd u ( i +1) 1 , . . . , u ( i +1) d +1 ∈ U i +1 \ U i +2 suc h that  v 1 , . . . , v d , . . . , u ( i ) 2 d +1 , . . . , v ( i ) 3 d | {z } T i , u ( i +1) 1 , . . . , u ( i +1) d  forms an extra-tight trail T ′ i +1 in G ′ . Note that b y (iii) , ∆( T ′ i +1 [ U i ]) ≤ 3 µ 2 | U i | . Let G ( i +1) : = G ′ \ T ′ i +1 . Th us, G ( i +1) is obtained from G by deleting the edge-sets of t wo extra-tight trails T ′ i +1 and T end . Since t wo of the ends of T ′ i +1 and T end are ( v 1 , . . . , v d ) and ( v ′ d , . . . , v ′ 1 ) , and the other t wo ends only hav e v ertices in U i +1 , the degree condition ( 4 ) on G implies that all vertices in U i \ U i +1 ha ve degree divisible b y d 2 in G ( i +1) . Hence, we can apply the Co ver Down Lemma 3.2 with G 3 . 2 : = G ( i +1) [ U i ] − G ( i +1) [ U i +2 ] , U 3 . 2 : = U i +1 , and α 3 . 2 : = 2 α . T o chec k the minimum degree condition of the Co ver Down Lemma, let S ∈  U i d − 1  . By prop ert y (iv) of an  α + µ 2 , µ, m  -v ortex, w e ha ve deg G ( S, U i ) ≥  1 − α − µ 2  | U i | . Therefore, deg G 3 . 2 ( S ) ≥  1 − α − µ 2  | U i | − ∆( T ′ i +1 [ U i ]) − ∆( T end [ U i ]) − | U i +2 | ≥  1 − α − µ 2  | U i | − 3 µ 2 | U i | − ∆( T end [ U i ]) − µ 2 | U i | ≥ (1 − 2 α ) | U i | . Hence, all conditions of the Cov er Down Lemma 3.2 are satisﬁed and we get u ( i +1) d +1 , . . . , u ( i +1) 3 d ∈ U i +1 together with an extra-tight trail T ′′ i +1 with ends  u ( i +1) d +1 , . . . , u ( i +1) 2 d  and  u ( i +1) 2 d +1 , . . . , u ( i +1) 3 d  suc h that G ′ \ G ′ [ U i +1 ] ⊆ T ′′ i +1 ⊆ G ′ \ G ′ [ U i +2 ] and ∆( T ′′ i +1 [ U i +1 ]) ≤ µ 2 | U i | . By Lemma 2.2 , we can ﬁnd vertices g ( i +1) 1 , . . . , g ( i +1) 2 d ∈ U i +1 \ U i +2 suc h that  v 1 , . . . , v d , . . . , u ( i +1) 1 , . . . , u ( i +1) d | {z } T ′ i +1 , g ( i +1) 1 , . . . , g ( i +1) 2 d , u ( i +1) d +1 , . . . , u ( i +1) 2 d , . . . , u ( i +1) 2 d +1 , . . . , u ( i +1) 3 d | {z } T ′′ i +1  forms an extra-tigh t trail T i +1 in G ′ whic h concludes the inductiv e step. Step 4. In the end, we ha ve an extra-tight trail T ℓ with ends ( v 1 , . . . , v d ) and  u ( ℓ ) 2 d +1 , . . . , u ( ℓ ) 3 d  where u ( ℓ ) 2 d +1 , . . . , u ( ℓ ) 3 d ∈ U ℓ , G ′ \ G ′ [ U ℓ ] ⊆ T ℓ ⊆ G ′ , and ∆( T ℓ [ U ℓ ]) ≤ 2 µ 2 | U ℓ | . Let G ′′ : = G ′ [ U ℓ ] \ T ℓ . It consists of m vertices and b y property (iv) of an  α + µ 2 , µ, m  -v ortex, we hav e δ ( G ′′ ) ≥  1 − α − µ 2  m − ∆( T ℓ [ U ℓ ]) ≥ (1 − 1 . 5 α ) m . Thus, Lemma 2.2 implies there exist 2 d vertices in U ℓ that connect the end  u ( ℓ ) 2 d +1 , . . . , u ( ℓ ) 3 d  of T ℓ to the end ( g ′ 1 , . . . , g ′ d ) of T end to form one long extra-tight trail T ℓ +1 with the ends ( v 1 , . . . , v d ) and ( v ′ d , . . . , v ′ 1 ) . But no w, the remaining graph L of uncov ered edges of G [ U ℓ ] fulﬁlls the conditions of the absorbing prop ert y of A . Hence, b y changing the v ertex sequence of A within T end within T ℓ +1 , we get an extra-tight trail that cov ers all the edges of G . 16 4 Absorb er Lemma The goal of this section is to prov e the Absorb er Lemma 3.3 , i.e. constructing an extra-tight trail A that can absorb the edges of any leftov er L ⊆ G [ U ] in to an extra-tight trail with the same ends. As indicated in Section 1.2 , w e dra w inspiration from the approac h of Glo ck, Joos, Kühn, and Osthus in [ 12 ], where they construct a tigh t d -uniform Euler tour in dense enough hypergraphs by ﬁrst constructing an appropriate “bac kb one” tight trail and then merging into it one by one the tight cycles of an appropriate decomp osition of the rest. If we w ere to follow [ 12 ] closely , we would just build the sequence of our extra-tight trail A in suc h a w ay that it con tains, disjointly , ev ery ( d − 1) -tuple ( u 1 , . . . , u d − 1 ) of distinct vertices from U . Then, for each cycle C of a decomp osition of some L in to extra-tight cycles, we would divert the sequence of A immediately after the app earance of some consecutive ( d − 1) -tuple of C , sending it ﬁrst to go around in C and only after that pro ceed further on A (see Figure 5 ). w 0 u 1 u 2 u d − 1 w d c 1 c 2 c k − 1 c k u 0 u d A C ⇝ w 0 u 1 u 2 u d − 1 w d c 1 c 2 c k − 1 c k u 0 u d Figure 5: Absorbing a cycle C into the trail A . V ertices that app ear in b oth A and C are shown in thic ker ellipses. The new sequence is ( . . . , w 0 , u 1 , u 2 , . . . , u d − 1 , u d , c 1 , . . . , c k , u 0 , u 1 , u 2 , . . . , u d − 1 , w d , . . . ) . This absorption approac h w orks like a charm in [ 12 ] for tigh t structures, but for extra-tight structures problems arise: the insertion of C into the sequence of A pro duces a sequence which co vers a set of edges a little bit diﬀerent from A ∪ C . Namely , the 2( d − 1) edges in E 1 = n { u 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , w d } , { w 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , u d } : i ∈ [ d − 1] o , (7) whic h were not cov ered here by A ∪ C , will b e cov ered here unnecessarily by the “rewired” sequence. On the other hand, those 2( d − 1) edges of A and C which jump ov er some vertex in ( u 1 , . . . , u d − 1 ) , i.e. those con tained in E 2 = n { w 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , w d } , { u 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , u d } : i ∈ [ d − 1] o , (8) are not co vered anymore by the new, rewired sequence (cf. Figure 5 ). T o o vercome this, we introduce our notion of an ( E 1 , E 2 ) -switc her, which is a pair of extra-tigh t trails whose edge set diﬀers exactly in E 1 and E 2 . Deﬁnition 4.1. Giv en d -graphs E 1 , E 2 , an ( E 1 , E 2 ) -switcher is a pair ( T 1 , T 2 ) of extra-tigh t trails with the same ends such that T 1 \ T 2 = E 1 , T 2 \ T 1 = E 2 and V ( E 1 ∪ E 2 ) is indep enden t in T 1 \ E 1 . The last prop erty , that V ( E 1 ∪ E 2 ) is indep endent in T 1 \ E 1 , will later b e helpful to ensure that no edge is rep eated in the ﬁnal absorber A and that U is indep endent in A . The plan is that our ev entual absorption trail A will also ha ve a segment where the sequence T 1 will b e con tained for every pair ( E 1 , E 2 ) sets whic h could arise at some cycle-insertion. If our absorption pro cedure for the cycles of a decomp osition of a particular L happened to use a cycle-insertion with corresp onding pair of edge sets ( E 1 , E 2 ) , then we also switch the corresp onding sequence T 1 in A to T 2 , th us undoing the wrong the cycle-insertion did to the set of edges co vered. The main result of the next subsections is that the desired switchers do exist. Lemma 4.2. L et d ≥ 2 , u 0 , . . . , u d , w 0 , w d b e d + 3 distinct vertic es. Then for the two families E 1 and E 2 of d -sets deﬁne d in ( 7 ) and ( 8 ), ther e exists an ( E 1 , E 2 ) -switcher ( T 1 , T 2 ) . F or the pro of of the ab ov e Switc her Lemma note that one can get from the family E 1 to the family E 2 b y switching the o ccurrences of the v ertices u 0 and w 0 , in other w ords E 1 ( u 0 ) = E 2 ( w 0 ) and E 1 ( w 0 ) = 17 E 2 ( u 0 ) . This motiv ates us to build an ( E 1 , E 2 ) -switc her ( T 1 , T 2 ) where one obtains the vertex sequence of the extra-tight trail T 2 from the vertex sequence of the extra-tight trail T 1 b y just sw apping all o ccurrences of the vertices u 0 and w 0 . Doing such a sw ap in any extra-tight trail creates another extra-tight trail and the status of an edge, in terms of whether it is cov ered, can only change if it contains exactly one of the sw app ed v ertices. W e will in fact make sure that T 1 do es not cov er an edge containing b oth u 0 and w 0 . The precise set of edges that c hanges its cov erage status is T 1 \ T 2 =  ( T 1 ( u 0 ) \ T 2 ( u 0 )) ∨ { u 0 }  ∪  ( T 1 ( w 0 ) \ T 2 ( w 0 )) ∨ { w 0 }  T 2 \ T 1 =  ( T 2 ( u 0 ) \ T 1 ( u 0 )) ∨ { u 0 }  ∪  ( T 2 ( w 0 ) \ T 1 ( w 0 )) ∨ { w 0 }  , where A ∨ B := { a ∪ b | a ∈ A, b ∈ B } . The sw ap causes the link graphs of u 0 and w 0 to swap, i.e. T 2 ( u 0 ) = T 1 ( w 0 ) and T 2 ( w 0 ) = T 1 ( u 0 ) . Thus, in order for T 1 and T 2 to satisfy the switc her prop erty , w e will need to make sure that the links of u 0 and w 0 in T 1 diﬀer in exactly the righ t w ay , namely that T 1 ( u 0 ) \ T 1 ( w 0 ) = E 1 ( u 0 ) = E 2 ( w 0 ) and T 1 ( w 0 ) \ T 1 ( u 0 ) = E 1 ( w 0 ) = E 2 ( u 0 ) . Since the construction of an ( E 1 , E 2 ) -switc her is quite complicated, as a w arm-up, in the next sub- section we motiv ate our approach in the simplest case d = 2 . This is only meant for the b eneﬁt of the exp osition, the reader is w elcome to skip o ver to the actual general pro of. 4.1 Motiv ating sk etc h for d = 2 The issue of inserting the sequence of an extra-tigh t cycle in to the sequence of an extra-tigh t trail is illustrated in Figure 6 for the case d = 2 . There, the black and red edges form an e xtra-tigh t trail ( . . . , w − 1 , w 0 , u 1 , w 2 , w 3 , . . . ) and an extra-tight cycle ( . . . , c k , u 0 , u 1 , u 2 , c 1 , . . . ) in tersecting in the ver- tex u 1 . If one attempted to merge the edges of these t wo into one extra-tight trail b y simply taking the v ertex-sequence ( . . . , w 0 , u 1 , u 2 , c 1 , . . . , c k , u 0 , u 1 , w 2 , . . . ) , then the set E 2 = { w 0 w 2 , u 0 u 2 } of red edges w ould no longer b e used, and the set E 1 = { u 0 w 2 , w 0 u 2 } of blue edges w ould b e used extra. w 0 w − 1 u 1 w 3 w 2 c k c 1 u 0 u 2 Figure 6: Merging the sequence of an extra-tight cycle into an extra-tight trail in the t wo-dimensional case. By our general contemplation, in order to construct the switcher, we need to make sure the ( 1 -uniform) links of u 0 and w 0 in T 1 diﬀer exactly the right wa y . T 1 ( u 0 ) \ T 1 ( w 0 ) = { w 2 } and T 1 ( w 0 ) \ T 1 ( u 0 ) = { u 2 } . Denoting these graphs b y ˜ G 1 := T 1 ( u 0 ) and ˜ G 2 := T 1 ( w 0 ) this means ˜ G 1 \ ˜ G 2 = { w 2 } and ˜ G 2 \ ˜ G 1 = { u 2 } . In the case d = 2 , this turns out to b e not that hard. F or example, T 1 = ( x 1 , w 2 , u 0 , x 2 , x 3 , w 0 , u 2 , x 1 ) , 18 w orks as ˜ G 1 = { x 1 , w 2 , x 2 , x 3 } and ˜ G 2 = { x 2 , x 3 , u 2 , x 1 } . With T 2 = ( x 1 , w 2 , w 0 , x 2 , x 3 , u 0 , u 2 , x 1 ) ob- tained by switching just w 0 and u 0 , ( T 1 , T 2 ) is indeed an ( E 1 , E 2 ) -switc her. Unfortunately , we w ere not able to ﬁnd a generalisation of this simple construction for larger d . T o- w ards the general case, as a simpler goal, w e now indicate ho w to build an extra-tight trail T where u 0 and w 0 ha ve the same link graph ˜ G , and hence sw apping all occurrences of u 0 and w 0 in the sequence pro duces an extra-tight trail co vering the very same set of edges. Note that eac h occurrence of a vertex v within the sequence of an extra-tight trail, which is not at one of the ends, results in four neighbours of v , spanning a path of three edges in the extra-tight trail. Namely , in the extra-tight trail ( . . . , v 1 , v 2 , v , v 3 , v 4 , . . . ) , the neigh b ours v 1 , v 2 , v 3 , v 4 of v span the edges v 1 v 2 , v 2 v 3 , v 3 v 4 in the extra-tight trail. F urthermore, diﬀerent o ccurrences of v in the sequence necessarily generate v ertex-disjoint paths. On the one hand, for our task, the v ertices occurring in the link of u 0 and w 0 ha ve to b e the same. On the other hand, as an extra-tigh t trail should cov er every 2 -set at most once, we ask the vertex-disjoin t paths of length three that app ear in the extra-tigh t trail around an o ccurrence of u 0 in the sequence to b e edge-disjoint from those that appear around an o ccurrence of w 0 in the sequence. While this is not necessary as the example ab o ve sho es, it will b e crucial for our construction of the switcher. This motiv ates us c ho osing 16 v ertices for the ( 1 -uniform) link ˜ G of u 0 and w 0 and aiming to co ver them by tw o sets of four paths of length three in an edge-disjoint fashion (see the blue and red paths on Figure 7 ). W e then create our sequence T of length 68 (cf. Figure 7 ) by inserting u 0 in the middle of eac h blue path and w 0 in the middle of each red path and connecting up these eigh t sequences of ﬁve v ertices with, say , four new padding vertices in b et ween them (see Lemma 2.2 ). The padding vertices are c hosen to be all distinct and their role is just to make sure that no pair of v ertices app ears more than once within three consecutive entries of the sequence. The k ey property of the sequence on Figure 7 is that swapping the o ccurrences of u 0 and w 0 do es not c hange the underlying graph. w 0 w 0 w 0 w 0 u 0 u 0 u 0 u 0 start end Figure 7: Extra-tigh t trail where w 0 and u 0 ha ve the same link ˜ G . The 1-edges of ˜ G are given by circles. Only those edges of the extra-tight trail are sho wn (in red, blue, or green) that do not contain a padding v ertex. Sw apping u 0 and w 0 in the indicated sequence of v ertices results in an extra-tight trail co vering the same edges. Recall, that in the actual pro of, the link graphs of u 0 and w 0 will not b e the same graph ˜ G , but rather w ell chosen ˜ G 1 and ˜ G 2 suc h that ˜ G 1 \ ˜ G 2 = { w 2 } and ˜ G 2 \ ˜ G 1 = { u 2 } . Then, w e still plan to cov er b oth 19 graphs with paths of order 4 where the corresponding 2-edges are distinct. 4.2 Constructing switc hers for arbitrary d F or general d , w e pro ceed analogously to the ab ov e, but the structures get more complex. Similarly to the ab ov e construction for d = 2 , we will need to get hold of some ( d − 1) -graphs ˜ G 1 and ˜ G 2 that can play the role of the link of u 0 and w 0 , and hence diﬀer in a very sp eciﬁc wa y . In an extra-tight path E P ( d ) 2 d +1 , the link of the middle v ertex, which w e think of b eing an occurrence of u 0 or w 0 within a long extra-tight trail, is an extra-tight path E P ( d − 1) 2 d of length one less. Hence, we will need to ﬁnd E P ( d − 1) 2 d -decomp ositions of ˜ G 1 and ˜ G 2 . Eac h copy of E P ( d − 1) 2 d is the shadow of some tigh t path P ( d ) 2 d , whic h are the red and blue edges ab o ve, and w e furthermore will ha ve to make sure that they are all distinct in the decomp ositions. T o ﬁnd the E P ( d − 1) 2 d -decomp ositions for the construction of switchers and the decomposition of the lefto ver L in the Absorb er Lemma into extra-tight cycles of constant length, w e will use the general theorem of [ 13 ] ab out F -decomp ositions. F or this, we need to introduce some terminology . Deﬁnition 4.3 ([ 13 ]) . F or a d -graph F , the divisibility ve ctor Deg( F ) : = ( g 0 , . . . , g d − 1 ) ∈ N d is giv en b y g i : = gcd  | F ( S ) | : S ∈  V ( F ) i  for i ∈ [ d − 1] 0 . In particular, g 0 = | F | and we write Deg( F ) i : = g i . Giv en tw o d -graphs F and G , G is called F -divisible if Deg( F ) i divides | G ( S ) | for all i ∈ [ d − 1] 0 and all S ∈  V ( G ) i  . An F -p acking in G is a collection of edge-disjoint copies of F in G . An F -packing F is an F -de c omp osition of G if every edge of G is contained in (exactly) one cop y of F in F . It is easy to see that G can only hav e an F -decomp osition if G is F -divisible. When G is a dense quasirandom d -graph, then the simple divisibility condition is also suﬃcient for an F -decomposition. W e use the follo wing con venien t notion of quasirandomness. Deﬁnition 4.4 ([ 18 ]) . A d -graph G on n v ertices is called ( c, h, p ) -typic al if for any family A of ( d − 1) - subsets of V ( G ) with | A | ≤ h , w e ha ve   T S ∈ A G ( S )   = (1 ± c ) p | A | n . F or us, ﬁnding an F -decomp osition is not enough. When we build the ( E 1 , E 2 ) -switc her, we hav e to ﬁnd t wo E P ( d − 1) 2 d -decomp ositions where certain d -sets are distinct (the red and blue edges in Figure 7 ). The following theorem enables us to do this. Theorem 4.5 ([ 13 ], Theorem 9.6) . L et 1 /n ≪ γ ≪ c, 1 /h ≪ p, 1 /f and r ∈ [ f − 1] . L et F b e any r -gr aph on f vertic es. Supp ose that G is a ( c, h, p ) -typic al F -divisible r -gr aph on n vertic es. L et O b e an ( r + 1) -gr aph on V ( G ) with ∆( O ) ≤ γ n . Then G has an F -de c omp osition F such that S F ∈F  V ( F ) r +1  and O ar e disjoint. W e will apply the previous theorem for the ( d − 1) -uniform extra-tight path E P ( d − 1) 2 d when building the ( E 1 , E 2 ) -switc her. In the pro of of the Absorb er Lemma, w e will also apply the theorem for the d -uniform extra-tight cycle E C ( d ) f . T o this end w e compute the divisibilit y v ector of these t wo graphs. Lemma 4.6. If d ≥ 2 and f ≥ d + 3 ar e ﬁxe d inte gers, then Deg  E C ( d ) f  i =      f · d i = 0 d 2 i = 1 1 else, for i ∈ [ d − 1] 0 and Deg  E P ( d − 1) 2 d  i = ( d 2 i = 0 1 else, for i ∈ [ d − 2] 0 . Pr o of. By Deﬁnition 1.4 , there is a 1-to-1 corresp ondence betw een the edges of E C ( d ) f and [ f ] × [ d ] . Therefore, the n umber of edges in E C ( d ) f is Deg  E C ( d ) f  0 = f · d . By Observ ation 2.1 , we ha ve that every vertex in E C ( d ) f has degree d 2 using that f ≥ d + 3 . Thus, we hav e Deg  E C ( d ) f  1 = d 2 . 20 Next, we will determine Deg  E C ( d ) f  i for some ﬁxed i ∈ [ d − 1] \{ 1 } . F or this, let ( v 1 , . . . , v f ) b e the vertex sequence of E C ( d ) f where we view the indices mo d f . W e will compute   E C ( d ) f ( S )   for S ∈  { v d , v d +1 , . . . , v d + i − 1 } , { v d − 1 , v d +1 , v d +2 , . . . , v d + i − 1 }  and show that these num bers are coprime. Starting with S = { v d , v d +1 , . . . , v d + i − 1 } , we determine which pairs ( ι, σ ) ∈ [ f ] × [ d ] corresp ond to an edge that con tains S . W e need that ι ∈ { i − 1 , i, . . . , d } and then w e ha ve to choose σ ∈ [ d ] such that v ι + σ ∈ S . There are d − i c hoices for σ unless ι = d in whic h case w e hav e d − i + 1 c hoices for σ . W e can conclude that   E C ( d ) f ( S )   = ( d − i + 2)( d − i ) + 1 = ( d − i + 1) 2 . Doing the same thing for S = { v d − 1 , v d +1 , v d +2 , . . . , v d + i − 1 } , w e see that we need ι ∈ { i − 1 , . . . , d − 1 } . Afterw ards, w e again hav e d − i c hoices for σ unless ι = d − 1 in whic h case w e ha ve d − i + 1 choices. Therefore,   E C ( d ) f ( S )   = ( d − i + 1)( d − i ) + 1 = ( d − i ) 2 + ( d − i ) + 1 . Because of gcd  ( d − i + 1) 2 , ( d − i ) 2 + ( d − i ) + 1  = gcd  ( d − i + 1) 2 , d − i  = 1 , w e can conclude Deg  E C ( d ) f  i = 1 . F or E P ( d − 1) 2 d , note that the edges of E P ( d − 1) 2 d are exactly the edges of a ( d − 1) -uniform extra-tight path with v ertex sequence ( v 1 , . . . , v 2 d ) . By Deﬁnition 1.4 , we hav e that E P ( d − 1) 2 d has  2 d − ( d − 1)  · ( d − 1) + 1 = d 2 edges. F urthermore, Observ ation 2.1 implies that v 1 has degree d − 1 whereas v 2 has degree 2 d − 3 . Th us, Deg  E P ( d − 1) 2 d  0 = d 2 and Deg  E P ( d − 1) 2 d  1 = 1 . F or i ∈ [ d − 2] \{ 1 } , w e can compute   E P ( d − 1) 2 d ( S )   for S ∈  { v d , v d +1 , . . . , v d + i − 1 } , { v d − 1 , v d +1 , v d +2 , . . . , v d + i − 1 }  . Then we get the same num b ers as for E C ( d ) f (but with d replaced b y d − 1 ). Thus, Deg  E P ( d − 1) 2 d  i = 1 . W e are now ready to build the ( E 1 , E 2 ) -switc her with the help of E P ( d − 1) 2 d . Pr o of of L emma 4.2 . W e wan t to apply Theorem 4.5 twice on carefully chosen ( d − 1) -graphs G 1 and G 2 whic h will give us tw o E P ( d − 1) 2 d -decomp ositions F and F ′ . W e apply Theorem 4.5 with r : = d − 1 , f : = 2 d , p : = 1 . Fix γ , c, 1 /h, 1 /n 0 > 0 small enough such that Theorem 4.5 holds for every n ≥ n 0 . Let V b e an n -element set suc h that u 1 , . . . , u d − 1 , u d , w d ∈ V , u 0 , w 0 / ∈ V and deﬁne ˜ G to b e the ( d − 1) -graph on vertex set V with edge set ˜ G =  V d − 1  \  { u 1 , . . . , u d , w d } d − 1  . Let ˜ G 1 b e the graph that is constructed from ˜ G by adding the edges in the set E 1 ( u 0 ) = E 2 ( w 0 ) =  { u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , w d } : i ∈ [ d − 1]  and then deleting an edge set E ′ ⊆  V \{ u 1 ,...,u d ,w d } d − 1  of size at most d 2 suc h that after the deletion of E ′ , the num b er of edges of ˜ G 1 is divisible by d 2 . Similarly , ˜ G 2 is constructed from ˜ G by adding the edges in E 1 ( w 0 ) = E 2 ( u 0 ) =  { u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , u d } : i ∈ [ d − 1]  and then deleting the same edge set E ′ . Note that the num b er of edges of ˜ G 2 is also divisible b y d 2 . Lemma 4.6 then implies that b oth ˜ G 1 and ˜ G 2 are E P ( d − 1) 2 d -divisible. Clearly , the complete ( d − 1) -graph is ( c, h, p ) -typical if n is large enough. Since ˜ G 1 and ˜ G 2 are b oth constructed from a complete ( d − 1) -graph by deleting at most    { u 1 ,...,u d ,w d , } d − 1    + | E ′ | ≤ ( d +1) · d 2 + d 2 edges, they are also ( c, h, p ) -typical provided n is large enough. Therefore, w e can ﬁrst apply Theorem 4.5 on ˜ G 1 , with the ( d − 1) -uniform extra-tight path E P ( d − 1) 2 d , and the empt y d -graph O . W e get an E P ( d − 1) 2 d -decomp osition F =  F ( d − 1) i  i ∈ [ k ] . Eac h F ( d − 1) i is the shado w of some d -uniform tight path, that we denote b y F ( d ) i . No w let O = S i ∈ [ k ] F ( d ) i . T o compute ∆( O ) , ﬁx a set S ⊆ V ( O ) of size d − 1 . Since S is an edge in at most one cop y of E P ( d − 1) 2 d in F , it can b e the subset of at most ∆  P ( d ) 2 d  = 2 edges of O . Th us, ∆( O ) ≤ 2 ≤ γ n for large enough n . W e can then apply Theorem 4.5 , but this time on ˜ G 2 to get another E P ( d − 1) 2 d -decomp osition F ′ =  F ′ ( d − 1) i  i ∈ [ k ] suc h that no d -set of O is con tained in any V  F ( d − 1) i  . Each F ′ ( d − 1) i is the shadow of some d -uniform tigh t path, that w e denote b y F ′ ( d ) i . The key prop erty of the decomp osition F ′ implies that no d -edge of some F ′ ( d ) j ′ app ears as a d -edge in some F ( d ) j . W e are now ready to build the switcher. W e will ﬁrst build the sequence T 1 and then obtain T 2 b y switc hing all o ccurrences of u 0 and w 0 . The sequence T 1 will b e the concatenation of sequences of length 21 2 d + 1 for each mem b er of the decomp ositions F and F ′ , together with 2 d individual “padding vertices” in b etw een them. Namely , if ( v 1 , . . . , v 2 d ) is the vertex sequence of a copy of E P ( d − 1) 2 d in F ∪ F ′ then w e add the v ertex sequence ( v 1 , . . . , v d , u 0 , v d +1 , . . . , v 2 d ) (9) if the cop y is in F and add the vertex sequence ( v 1 , . . . , v d , w 0 , v d +1 , . . . , v 2 d ) (10) if the cop y is in F ′ . F urthermore, b etw een these vertex sequences, with Lemma 2.2 w e alwa ys add 2 d en tirely new vertices that ha ve not app eared b efore. W e call these vertices the unique v ertices of our sequence. Let T 1 b e the resulting sequence of length 2 k (4 d + 1) − 2 d . T o see that T 1 indeed deﬁnes an extra-tigh t trail, w e ﬁrst note that every in terv al of length 2 d + 1 in T consists of distinct v ertices as the sequences of b oth ( 9 ) and ( 10 ) come from paths and a new vertex and 2 d unique vertices are inserted in b et ween them. T o ha ve an extra-tigh t trail w e need that no d -set e app ears m ultiple times among d + 1 consecutive elements of T 1 . If e contains a unique vertex v , then all app earances of e within d + 1 consecutive elements of T must o ccur within the in terv al S of 2 d + 1 consecutive elements of T having v at its midp oint. W e ha ve seen ab o ve that this in terv al consists of distinct vertices so the placement of e is unique. If e do es not con tain an y unique v ertex, then eac h o ccurrence of it within d + 1 consecutiv e elemen ts of T must o ccur within one of the subsequences of type ( 9 ) or t yp e ( 10 ). If furthermore u 0 ∈ e , then e \ { u 0 } is a ( d − 1) -edge in ˜ G 1 whic h app ears in exactly one of the copies F ( d − 1) j of E P ( d − 1) 2 d in the decomp osition F , hence the placemen t of e is also unique. If instead w 0 ∈ e then e \ { w 0 } is a ( d − 1) -edge in ˜ G 2 whic h app ears in exactly one of the copies F ′ ( d − 1) j of E P ( d − 1) 2 d in the decomp osition F ′ , hence the placemen t of e is again unique. Finally , if u 0 , w 0 ∈ e , then e is contained in either F ( d ) j or F ′ ( d ) j for some unique j ∈ [ k ] . Here the F ( d ) i are all disjoint from each other since the ( d − 1) -subsets of their members are distinct due to F b eing a decomp osition. Analogously , the F ′ ( d ) j are also all disjoint from eac h other since F ′ is a decomp osition. Finally , any F ( d ) j and F ′ ( d ) j ′ are also disjoin t as in the creation of F ′ , w e required that no d -subset of the vertex sets V  F ( d ) j  is con tained in any of the vertex sets V  F ′ ( d ) j ′  . W e can thus conclude that the sequence T indeed induces an extra-tigh t trail. Next, we prov e that our extra-tight trail T 1 co vers all edges of E 1 . F or any j ∈ [ d − 1] , consider the d -set e = { u 0 , u 1 , . . . , u j − 1 , u j +1 , . . . , u d − 1 , w d } ∈ E 1 . Note that e \{ u 0 } is in ˜ G 1 . As the link of u 0 in the extra-tigh t path of the sequence ( 9 ) is exactly F ( d − 1) i and since the F ( d − 1) i form a decomposition of ˜ G 1 , we can conclude that the link of u 0 in our extra-tigh t trail is exactly ˜ G 1 . Hence, e is cov ered b y the extra-tight trail. An analogous argumen t shows that the link of w 0 in our extra-tight trail is exactly ˜ G 2 and that the d -set { w 0 , u 1 , . . . , u j − 1 , u j +1 , . . . , u d − 1 , u d } of E 1 is cov ered by the extra-tight path of a sequence of ( 10 ). Hence, the full E 1 is cov ered. Next, we will show that V ( E 1 ∪ E 2 ) is an indep endent set in T 1 \ E 1 . Let e ∈  { u 0 ,...,u d ,w 0 ,w d } d  . If u 0 ∈ e , w e can use that the link of u 0 in T 1 is exactly ˜ G 1 . By deﬁnition of ˜ G 1 , w e hav e ˜ G ∩  { u 1 ,...,u d ,w 0 ,w d } d  = E 1 . Hence, e ∈ T 1 \ E 1 . An analogous argument works if w 0 ∈ e . Th us, we we can assume that e ∈  { u 1 ,...,u d ,w d } d  . W e will show that e cannot b e cov ered b y the extra-tight trail. Supp ose it is. Then it m ust already b e cov ered by one of the sequences of ( 9 ) or ( 10 ). Without loss of generalit y , assume it is a sequence of ( 9 ). But then every ( d − 1) -subset of e is in the link of u 0 . How ever, the link of u 0 in T 1 is ˜ G 1 whic h only contains d − 1 edges of  { u 0 ,...,u d ,w 0 ,w d } d − 1  . As the num ber of ( d − 1) -subsets of e is d , this is a con tradiction. Finally , to obtain the extra-tight trail T 2 with the same ends as T 1 w e use the exact same v ertex sequence but sw ap all u 0 and w 0 . Then, as we only swapped the role of tw o v ertices, the v ertex-sequence indeed still deﬁnes an extra-tigh t trail. T o see that T 2 no w co vers exactly ( T 1 \ E 1 ) ∪ E 2 , note that only the edges con taining u 0 or w 0 are aﬀected when u 0 and w 0 are sw app ed. Before the swap, the link of u 0 w as ˜ G 1 , whereas it is ˜ G 2 after the sw ap. F or w 0 , it is the other w ay around. Since, by construction, ˜ G 1 \ ˜ G 2 = E 1 ( u 0 ) = E 2 ( w 0 ) and ˜ G 2 \ ˜ G 1 = E 1 ( w 0 ) = E 2 ( u 0 ) , the swap only causes that E 1 is no longer co vered, whereas E 2 is. As neither u 0 nor w 0 app ears at an end of the extra-tigh t trail, the ends of the extra-tigh t trail are the same before and after the swap. 22 4.3 Pro of of the Absorb er Lemma Recall that the plan is to construct the extra-tight trail A suc h that for each extra-tight cycle C in the left-o ver L ⊆ G [ U ] , A can b e rewired to path through C , while simultaneously containing an appropriate switc her that, when activ ated, rev erts all the eﬀects of the rewiring. Hence, for any sequence S = ( u 0 , u 1 , . . . , u d ) potentially contained in C , we hav e to include the sequence D S = ( w 0 , u 1 , . . . , u d − 1 , w d ) for some w 0 , w d ∈ V ( G ) \ U as well as the sequence T 1 from an ( E 1 , E 2 ) -switc her ( T 1 , T 2 ) in our absorption trail A . That wa y , if some S is used to absorb a cycle C , then at D S w e divert the sequence of A through C and swap the corresp onding subsequence T 1 of A to T 2 , with only the edges of C cov ered additionally . T o mak e it p ossible to ﬁnd such an extra-tight trail A in G , it is crucial that the n umber of potential attac hment tuples and the length of the trails T 1 are a function of m and d , while 1 /n ≪ 1 /m, 1 /d . Pr o of of L emma 3.3 . Let 1 /n ≪ η ≪ 1 /m ≪ α ≪ 1 /d ≤ 1 / 2 . Let S b e the set of all ordered tuples S = ( u 0 , u 1 , . . . , u d − 1 , u d ) of d + 1 distinct vertices of U , suc h that all edges of the corresp onding extra-tight path are in G . F or each S = ( u 0 , u 1 , . . . , u d ) ∈ S , we w ant to ﬁnd tw o v ertices w S 0 and w S d in V ( G ) \ U and an embedding of T S 1 ∪ T S 2 in G where  T S 1 , T S 2  is the  E S 1 , E S 2  -switc her giv en b y Lemma 4.2 with E S 1 = n { u 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , w S d } , { w S 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , u d } : i ∈ [ d − 1] o E S 2 = n { w S 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , w S d } , { u 0 , u 1 , . . . , u i − 1 , u i +1 , . . . , u d − 1 , u d } : i ∈ [ d − 1] o . F urthermore, we wan t to make sure that V  T S 1 ∪ T S 2  ∩ U = S and V  T S 1 ∪ T S 2  ∩ V  T S ′ 1 ∪ T S ′ 2  = S ∩ S ′ for any other tuple S ′ ∈ S where S ∩ S ′ is the set of all vertices that app ear in b oth tuples S and S ′ . Em b edding all T S 1 ∪ T S 2 in G can be done greedily , b y going through all S ∈ S one by one and em b edding T S 1 ∪ T S 2 one v ertex at a time, starting with the vertices of S which are already in G . Ev ery time w e embed the next vertex of T S 1 ∪ T S 2 in V \ U , we just hav e to mak e sure that all the necessary edges are present in G and that the v ertex c hosen do es not app ear in the embedding of any of the other T S ′ 1 ∪ T S ′ 2 . As   V  T S 1 ∪ T S 2    only dep ends on d and |S | only dep ends on m and d , we can alw ays ﬁnd a suitable next vertex by the minimum degree condition δ ( G ) ≥ (1 − α ) n , provided α <    V  T S 1 ∪ T S 2    d − 1  − 1 and n is c hosen large enough. Let D S b e the extra-tigh t trail  w S 0 , u 1 , . . . , u d − 1 , w S d  . W e connect all the D S and T S 1 to one long v ertex sequence b y iteratively inserting 2 d en tirely new gluing vertices of V \ U b et ween them using Lemma 2.2 . This, and that all new vertices can b e chosen distinct, is again possible b y the minim um degree condition δ ( G ) ≥ (1 − α ) n . Finally , we similarly add d entirely new vertices w 1 , . . . , w d of V \ U at the b eginning and d entirely new vertices w ′ d , . . . , w ′ 1 of V \ U at the end. Let A b e the resulting sequence and note that we hav e made sure that ev ery edge of A do es app ear in G . W e still hav e to see ho wev er that A induces an extra-tight trail, that is no d -subset e app ears multiple times among d + 1 consecutive elements of A . If e con tains one of the gluing v ertices, then e does not app ear multiple times by the same reason as in the previous lemma. If e do es not contain a gluing v ertex, then it is an edge in the extra-tight trail induced b y D S or T S 1 for some S ∈ S . First, we argue why e ∈ T S ′ 1 ∪ D S ′ for any S ′ ∈ S \ { S } . This is b ecause by construction V ( D S ) ∩ V ( D S ′ ) , V  T S 1  ∩ V  D S ′  , V ( D S ) ∩ V  T S ′ 1  , V  T S 1  ∩ V  T S ′ 1  are all contained in S ∩ S ′ ⊆ U and no edge of D S and T S 1 is entirely in U . Thus, we only ha ve to chec k that e do es not app ear in both D S and T S 1 . Here we use that, b y the deﬁnition of a switcher, V ( D S ) ⊆ V  E S 1 ∪ E S 2  is an indep enden t set in T S 1 \ E S 1 . F urthermore, D S ∩ E S 1 = ∅ . Thus, we can conclude that A induces an extra-tigh t trail. Next, we chec k that A has all the desired properties stated in the Absorb er Lemma. Note that b y the same argument as ab ov e, A do es not contain an y edge that lies entirely in U . F urthermore, we can mak e sure that ∆( A ) ≤ η n , as even the size of A dep ends only on d and m . T o chec k the ﬁnal property , let L by any d -graph on U with δ ( L ) ≥ (1 − α ) n where each 1-degree is divisible by d 2 . W e hav e to show that there is an extra-tigh t trail with edge set L ∪ A whose ends are the same as that of A . W e ﬁrst show this in the case when L can b e partitioned in to extra-tight cycles of length at least d + 3 b y absorbing the edge set of each of them into A . Let C be one of these extra-tight cycles and let u 0 , . . . , u d b e d + 1 consecutiv e vertices of C in that order. In other w ords, C is deﬁned by some cyclic sequence u 0 , . . . , u d , c 1 , . . . , c k . Let S = ( u 0 , . . . , u d ) . By construction, A contains the vertex sequence 23 D S , i.e. w S 0 , u 1 , . . . , u d − 1 , w S d . W e replace this in A by w S 0 , u 1 , . . . , u d − 1 , u d , c 1 , . . . , c k , u 0 , u 1 , . . . , u d − 1 , w S d and remov e C from L . By this change, the edges of E S 2 are no longer cov ered in A ∪ L , whereas the edges of E S 1 are now co vered twice since they were already cov ered b y T S 1 . But this is exactly what the ( E S 1 , E S 2 ) -switc her ( T S 1 , T S 2 ) was created for: b y replacing the v ertex sequence T S 1 in A by T S 2 , we can ensure that the same edges are cov ered as b efore. Therefore, this pro cedure can iteratively absorb each extra-tight cycle of the decomp osition of L , until we end up with an extra-tight trail cov ering exactly the edges A ∪ L while still ha ving the same ends ( w 1 , . . . , w d ) and ( w ′ d , . . . , w ′ 1 ) as A . It remains to show that L can indeed alwa ys b e decomp osed into extra-tigh t cycles. T o use Theo- rem 4.5 , w e hav e to make sure that L is E C ( d ) f -divisible for some ﬁxed f . F or conv enience, here w e will aim for f = 3 d . By Lemma 4.6 and b y the fact that all degrees of L are divisible b y d 2 , we only hav e to ensure that the num ber of edges is divisible by 3 d 2 . This is not necessarily the case to begin with, but since all degrees of L are divisible by d 2 , the d -uniform Handshak e Lemma at least implies that | L | is at divisible by d . Hence, there is an in teger k ∈ { 0 , . . . , 3 d − 1 } suc h that | L | ≡ kd (mo d 3 d 2 ) . Our strategy of partitioning L in to extra-tigh t cycles then starts b y ﬁnding a copy of E C ( d ) 3 d + k in L . Remo ving it lea ves us with a graph L ′ satisfying | L ′ | ≡ 0 (mo d 3 d 2 ) as | E C ( d ) 3 d + k | = (3 d + k ) d b y Obser- v ation 2.1 . Hence L ′ is E C ( d ) 3 d -divisible, b ecause deleting a tigh t cycle do es not change the divisibilit y of the vertex-degrees by d 2 . The extra-tigh t cycle of length 3 d + k can be found in L vertex by v ertex: when c ho osing the next v ertex, one only has to ensure that all the at most d 2 edges that are formed with the already c hosen v ertices in the extra-tight cycle are presen t in L . Eac h edge can exclude at most αm vertices. Thus, if α < 1 d 2 and m is large enough, then there is alw ays at least one vertex in U that can b e chosen next. After we remov ed th e extra-tight cycle of length 3 d + k , the remaining graph L ′ still satisﬁes δ ( L ′ ) ≥ (1 − α ) m − 4 ≥ (1 − 2 α ) m . As L ′ is E C ( d ) 3 d -divisible we apply Theorem 4.5 with O : = ∅ to ﬁnd the desired E C ( d ) 3 d -decomp osition of the graph L ′ . T o conclude, we recapitulate the order in whic h the parameters are chosen for the en tire pro of to w ork. Once d is ﬁxed, the size of each switc her  T S 1 , T S 2  is determined. Theorem 4.5 with parameters f : = 3 d , r : = d , p : = 1 deﬁnes parameters c , h , γ and n 0 suc h that the statement of Theorem 4.5 holds for all n 4 . 5 ≥ n 0 . Afterw ards, w e ﬁx α small enough such that we can greedily build all switchers in G ( α <  | V ( T S 1 ∪ T S 2 ) | d − 1  − 1 ), ﬁnd all gluing v ertices ( α < d − 2 ) and make sure that each graph L ′ on m vertices with δ ( L ′ ) ≥ (1 − 2 α ) m is ( c, h, 1) -typical ( α ≤ c 2 h ) so that w e can actually apply Theorem 4.5 for L ′ with E C ( d ) 3 d . Next, w e c ho ose m suc h that it is at least n 0 and that w e can build the extra-tigh t cycle of length k ≤ 3 d greedily in any graph L on m vertices with minimum degree δ ( L ) ≥ (1 − α ) m . Finally , we pic k any p ositive num ber for η and choose n large enough such that all switchers and gluing vertices can b e found greedily in an y graph G on n vertices with δ ( G ) ≥ (1 − α ) n and suc h that the maximum degree of the ﬁnal absorb er, whic h only dep ends on d and m , is at most η n . 5 Ro oted Em b eddings A t se v eral places, w e will ha ve to ﬁnd em b eddings of a ﬁxed graph T into G where the images of sev eral v ertices of T are already determined. F or example, w e already used that in the pro of of Theorem 1.8 when w e greedily found d v ertices suc h that they together with some other already de termined vertices form an extra-tigh t trail. These embeddings are called “ro oted em b eddings” and their existence has already b een studied in [ 13 ]. F or completeness, we rep eat their deﬁnitions here. Let T b e a d -graph and X ⊆ V ( T ) . A r o ot of ( T , X ) is a set S ⊆ X suc h that | S | ∈ [ d − 1] and | T ( S ) | > 0 . A G -lab el ling of ( T , X ) is an injectiv e map Λ : X → V ( G ) and, giv en a G -lab elling Λ , we sa y that an injective homomorphism ϕ : T → G is a Λ -faithful emb e dding of ( T , X ) into G if ϕ | X = Λ . F urthermore, a G -labelling Λ of ( T , X ) r o ots a v ertex set S ⊆ V ( G ) if S ⊆ Im(Λ) and   T  Λ − 1 ( S )    > 0 . Finally , the de gener acy of T r o ote d at X is the smallest D suc h that there is an ordering v 1 , . . . , v k of the elemen ts of V ( T ) \ X such that for every ℓ ∈ [ k ] , we hav e | T [ X ∪ { v 1 , . . . , v ℓ } ]( v ℓ ) | ≤ D . W e will need to ﬁnd sev eral ro oted embeddings sim ultaneously in the same graph G such that the images are edge-disjoint. The following do es that and is a slight v ariation of [ 13 , Lemma 5.20]. On one hand, it is less general since it remo ves restrictions enco ded by a d + 1 -graph O and the concept of 24 a hul l of ( T , X ) , which strengthens our condition (i) b elow. On the other hand, it has the additional requiremen t that all v ertices v ∈ V ( T ) \ X are em b edded in a sp ecial v ertex set U ⊆ V ( G ) . Lemma 5.1. L et 1 /n ≪ γ ≪ γ ′ ≪ ξ , 1 /t, 1 /D and d ∈ [ t ] and m ≤ γ n d b e an inte ger. F or every j ∈ [ m ] , let T j b e a d -gr aph on at most t vertic es and X j ⊆ V ( T j ) such that T j [ X j ] is empty and T j has de gener acy at most D r o ote d at X j . L et G b e a d -gr aph on n vertic es and U ⊆ V ( G ) such that for al l A ⊆  V ( G ) d − 1  with | A | ≤ D , we have   T S ∈ A G ( S ) ∩ U   ≥ ξ n . F or every j ∈ [ m ] , let Λ j b e a G -lab el ling of ( T j , X j ) . Supp ose that for al l S ⊆ V ( G ) with | S | ∈ [ d − 1] , we have that |{ j ∈ [ m ] : Λ j r o ots S }| ≤ γ n d −| S | − 1 . (11) Then for every j ∈ [ m ] , ther e exists a Λ j -faithful emb e dding ϕ j of ( T j , X j ) into G such that the fol lowing hold: (i) for al l distinct j, j ′ ∈ [ m ] , the d -gr aphs ϕ j ( T j ) and ϕ j ′ ( T j ′ ) ar e e dge-disjoint; (ii) ∆( S j ∈ [ m ] ϕ j ( T j )) ≤ γ ′ n ; (iii) for al l j ∈ [ m ] and al l v ∈ V ( T j ) \ X j we have ϕ j ( v ) ∈ U . The pro of of this lemma is almost identical to the pro of of [ 13 , Lemma 5.20], hence, we omit it here. W e will mainly use this lemma to connect a given set of extra-tight trails into one large extra-tight trail. This is giv en b y the following lemma: Lemma 5.2. L et 1 /n ≪ γ ≪ γ ′ ≪ ξ , 1 /D ≪ 1 /d . Supp ose that G is a d -gr aph on n vertic es and let U ⊆ V ( G ) such that for al l A ⊆  V ( G ) d − 1  with | A | ≤ D , we have   T S ∈ A G ( S ) ∩ U   ≥ ξ n . L et P b e a c ol le ction of p airwise e dge-disjoint extr a-tight p aths such that e ach S ⊆ V ( G ) of size d − 1 app e ars at an end of at most γ n of the p aths in P and S P ∈P P ∩ G = ∅ . Then ther e is a subset H ⊆ G such that H ∪ S P ∈P P is the e dge set of an extr a-tight tr ail whose ﬁrst and last d vertic es al l lie in U and ∆( H ) ≤ γ ′ n . Pr o of. Let P = { P 1 , . . . , P m } and  u ( j ) 1 , . . . , u ( j ) d  resp.  w ( j ) d , . . . , w ( j ) 1  b e the ﬁrst resp. last d v ertices of P j where j ∈ [ m ] . Since eac h S ⊆ V ( G ) of size d − 1 app ears at an end of at most γ n of the paths in P , w e hav e m = |P | ≤ γ n d . F urther, let P 0 =  u (0) 1 , . . . , u (0) d  and P m +1 =  w ( m +1) d , . . . , w ( m +1) 1  b e tw o extra-tigh t paths giv en by t wo arbitrary edges from G [ U ] . In the end, these vertices will form the ends of our extra-tigh t trail. Let T b e the extra-tight path with vertex sequence ( w d , w d − 1 , . . . , w 1 , v 1 , . . . , v d , u 1 , . . . , u d ) but which only con tains those d -edges that hav e non-empt y intersection with { v 1 , . . . , v d } . The idea is to embed a cop y of T into G for each j ∈ [ m ] 0 suc h that w i is mapp ed to w ( j ) i and u i is mapp ed to u ( j +1) i for each i ∈ [ d ] . Then P j , the em b edded cop y of T , and P j +1 form one long extra-tight trail. T o mak e this more precise, we apply Lemma 5.1 . F or j ∈ [ m ] 0 , deﬁne T j : = T and X j : = { w d , . . . , w 1 , u 1 , . . . , u d } . W e c ho ose D depending on d large enough such that T j has degeneracy at most D rooted at X j . By assumption, G 5 . 1 : = G −  u (0) 1 , . . . , u (0) d  −  w ( m +1) 1 , . . . , w ( m +1) d  satisﬁes the requiremen ts for Lemma 5.1 with ξ 5 . 1 : = ξ / 2 . F or j ∈ [ m ] 0 , let Λ j b e the G 5 . 1 -lab elling of ( T j , X j ) that maps w i to w ( j ) i and u i to u ( j +1) i for each i ∈ [ d ] . T o chec k the last condition of Lemma 5.1 , let S ⊆ V ( G ) with | S | ∈ [ d − 1] . Since the extra- tigh t trail T has d vertices betw een w 1 and u 1 , the only j ∈ [ m ] such that Λ j ro ots S are those where S ⊆  w ( j ) d , . . . , w ( j ) 1  or S ⊆  u ( j +1) 1 , . . . , u ( j +1) d  . This happ ens for at most 2 γ n d −| S | of the j ∈ [ m ] . Hence, we can apply Lemma 5.1 with γ 5 . 1 : = 3 γ . W e get edge-disjoin t, Λ j -faithful em- b eddings ϕ j of ( T j , X j ) such that ∆( S j ∈ [ m ] 0 ϕ j ( T j )) ≤ γ ′ n . By construction, H : = S j ∈ [ m ] 0 ϕ j ( T j ) ∪  { u (0) 1 , . . . , u (0) d } , { w ( m +1) 1 , . . . , w ( m +1) d }  has the desired prop erties. 6 Appro ximate Decomp osition Lemma The goal of this section is to prov e the Approximate Decomp osition Lemma 3.1 which we will need in the pro of of the Co ver Down Lemma. W e emplo y a tec hnique that w as recen tly used b y Gish b oliner, Glo ck, and Sgueglia [ 11 ] in a diﬀerent setting, namely in a w ork on tigh t Hamilton cycles. Roughly sp eaking, the 25 idea is that if one wan ts to ﬁnd a long “path–like” structure, a natural approach is to use a random walk argumen t whic h extends the structure via randomized steps. How ever, if the structure one needs to ﬁnd is v ery long, the random pro cedure needs guidance to av oid forbidden self-intersections. This can make the analysis quite in tricate. Instead, the approach used in [ 11 ] is to sample many constan t-length paths using the same simple random w alk distribution, where self-intersection is not an issue. The obtained paths then form an auxiliary hypergraph in which one can ﬁnd an almost-perfect matching using standard nibble results. Dep ending on the speciﬁc setup, the matc hing condition corresp onds then to av oiding forbidden in tersections b etw een the diﬀerent paths. Of course, the result of this is not one long structure, but rather many disjoint pieces. So in the ﬁnal step, one needs to stitch these individual pieces together. Since we lo ok at extra-tight trails instead of tight Hamilton cycles, our pro of of the Approximate Decomp osition Lemma has some diﬀerences: we wan t to co ver almost all edges instead of almost all v ertices, which is why our auxiliary hypergraph will b e deﬁ ned diﬀerently , and w e ha ve to analyze the probabilit y that an edge is co vered by a randomly sampled constan t-length path. F urthermore, the ﬁnal step of stitc hing the individual pieces together is a bit diﬀerent, b ecause w e also need to make sure that no edge app ears to o often at an end of these paths. Also, since we consider extra-tight paths, we cannot use p erfect fractional matc hings to deﬁne the suitable probability for the random w alk, but we ha ve to use fractional K d +1 -decomp ositions. This is what w e deﬁne ﬁrst. Deﬁnition 6.1. F or a d -graph G , let K d +1 ( G ) b e the set of all ( d + 1) -subsets S of V ( G ) such that G [ S ] is a complete d -graph. A fr actional K d +1 -de c omp osition of G is a function x : K d +1 ( G ) → [0 , 1] such that for every e ∈ G , we hav e X S ∈ K d +1 ( G ) e ⊆ S x ( S ) = 1 . F or µ ∈ (0 , 1] , a fractional K d +1 -decomp osition x is µ -normal if µn − 1 ≤ x ( S ) ≤ µ − 1 n − 1 for all S ∈ K d +1 ( G ) . W e need make sure that we hav e a suﬃciently normal fractional K d +1 -decomp osition in our graph G : Lemma 6.2. L et 1 /n ≪ α ≪ 1 /d ≤ 1 / 2 . L et G b e an n -vertex d -gr aph with δ ( G ) ≥ (1 − α ) n . Then G has a 0 . 9 -normal fr actional K d +1 -de c omp osition. In [ 2 , Theorem 1.5], it is shown that G indeed has a fractional K d +1 -decomp osition b ecause α ≪ 1 /d . Unfortunately , the theorem does not mak e an y commen ts on the µ -normalit y of the decomp osition. Ho wev er, a closer examination of the pro of of [ 2 , Theorem 1.5] reveals that their decomp osition is in fact µ -normal if α is small enough. Since the pro of is v erbatim the same, w e only sketc h it. Pr o of. The proof of [ 2 , Theorem 1.5] w orks as follows: First, each K d +1 is assigned the w eight ω : = 1 /κ where κ : = P e ∈ G κ ( d +1) e / | G | and κ ( d +1) e ∈ [(1 − dα ) n, n ] is the num b er of copies of K ( d ) d +1 that contain e . Hence, at the b eginning eac h K d +1 is assigned the weigh t 1 /κ ∈ [ n − 1 , 1 1 − dα n − 1 ] . Ho wev er, this is not necessarily a fractional K d +1 -decomp osition. Therefore, the authors change the w eights slightly to obtain a fractional K d +1 -decomp osition. They show that the weigh t of eac h cop y of K ( d ) d +1 is changed b y at most 2 d +2 d 2 ( d +1) 2 d − 1 αω d ! P d j =0 1 2 j j ! . Hence, for ﬁxed d and small enough α , we ha ve a 0 . 9 -normal fractional K d +1 -decomp osition. No w that w e ha ve established that a fractional K d +1 -decomp osition x : K d +1 ( G ) → [0 , 1] exists, w e will describ e ho w this can b e used to deﬁne a probabilit y distribution. First, we extend x to a function  V ( G ) d +1  → [0 , 1] by setting x ( S ) : = 0 for all S ∈  V ( G ) d +1  \ K d +1 ( G ) . W e deﬁne the following random walk Y = ( Y 1 , Y 2 , . . . ) in V ( G ) . Let ( V ) d b e the set of all ordered d -tuples of V ( G ) whose unordered set forms an edge in G . The ﬁrst d vertices ( Y 1 , . . . , Y d ) are c hosen uniformly at random among ( V ) d . F or i ≥ d , let − → Z i : = ( Y i − ( d − 1) , . . . , Y i ) b e the ordered set of the last d v ertices of the w alk and let Z i : = { Y i − ( d − 1) , . . . , Y i } be the corresp onding set. F or an y v ∈ V ( G ) \ Z i , we deﬁne the transition probabilit y as follows: Pr[ Y i +1 = v | Y i − ( d − 1) , . . . , Y i ] = x ( Z i ∪ { v } ) . (12) 26 F or v ∈ Z i , the probabilit y Pr[ Y i +1 = v | Y i − ( d − 1) , . . . , Y i ] is set to 0. Note that this indeed deﬁnes a probabilit y distribution b ecause X v ∈ V ( G ) \ Z i x ( Z i ∪ { v } ) = X S ∈ K d +1 ( G ) Z i ⊆ S x ( S ) = 1 . Here, we use the fact that Z i is alwa ys an edge of G . Note that Y is equiv alent to the random w alk Z =  − → Z d , − − − → Z d +1 , . . .  . Here, Z is a Marko v chain with state space ( V ) d . F urthermore, if we deﬁne π  − → S  : = 1 | ( V ) d | for all − → S ∈ ( V ) d , then π deﬁnes a stationary distribution of the Marko v chain. Indeed, let P  − → S , − → T  b e the transition probability of − → S to − → T for any − → S , − → T ∈ ( V ) d . F or any ﬁxed − → T = ( t 1 , . . . , t d ) ∈ ( V ) d , we then get X − → S ∈ ( V ) d π  − → S  P  − → S , − → T  = 1 | ( V ) d | X − → S ∈ ( V ) d P  − → S , − → T  = 1 | ( V ) d | X v ∈ V ( G ) \{ t 1 ,...,t d − 1 } P  ( v , t 1 , . . . , t d − 1 ) , ( t 1 , . . . , t d )  = 1 | ( V ) d | X v ∈ V ( G ) \{ t 1 ,...,t d − 1 } x ( { v , t 1 , . . . , t d } ) = 1 | ( V ) d | X S ∈ K d +1 ( G ) { t 1 ,...,t d }⊆ S x ( S ) = 1 | ( V ) d | = π ( − → T ) . The following lemma shows that if j 1 , . . . , j d are d out of d +1 consecutive num b ers, then the probability that ( Y j 1 , . . . , Y j d ) forms a ﬁxed elemen t of ( V ) d is 1 | ( V ) d | . Lemma 6.3. L et i ≥ 1 b e ﬁxe d and let i = j 1 < j 2 < · · · < j d ≤ i + d . Fix an − → S = ( s 1 , . . . , s d ) ∈ ( V ) d . Then Pr[ Y j 1 = s 1 , . . . , Y j d = s d ] = 1 | ( V ) d | . Pr o of. If { j 1 , . . . , j d } = { i, i + 1 , . . . , i + d − 1 } = Z i + d − 1 , then Pr[ Y j 1 = s 1 , . . . , Y j d = s d ] = Pr  − − − − → Z i + d − 1 = − → S  = 1 | ( V ) d | where the last equality comes from the fact that the uniform distribution is stationary for the Marko v chain  − → Z d , − − − → Z d +1 , . . .  . Hence, w e can assume that j 1 = i, . . . , j ℓ = i + ℓ − 1 , j ℓ +1 = i + ℓ + 1 , . . . , j d = i + d for some 1 ≤ ℓ ≤ d − 1 . Let E b e the ev ent that Y j 1 = s 1 , . . . , Y j d = s d . Then Pr[ E ] = X v ∈ V ( G ) Pr h E | − − − − → Z i + d − 1 = ( s 1 , . . . , s ℓ , v , s ℓ +1 , . . . , s d − 1 ) i · Pr h − − − − → Z i + d − 1 = ( s 1 , . . . , s ℓ , v , s ℓ +1 , . . . , s d − 1 ) i = X v ∈ V ( G ) { s 1 ,...,s d ,v }∈ G Pr h E | − − − − → Z i + d − 1 = ( s 1 , . . . , s ℓ , v , s ℓ +1 , . . . , s d − 1 ) i · Pr h − − − − → Z i + d − 1 = ( s 1 , . . . , s ℓ , v , s ℓ +1 , . . . , s d − 1 ) i = 1 | ( V ) d | X v ∈ V ( G ) { s 1 ,...,s d ,v }∈ G Pr h E | − − − − → Z i + d − 1 = ( s 1 , . . . , s ℓ , v , s ℓ +1 , . . . , s d − 1 ) i = 1 | ( V ) d | X v ∈ V ( G ) { s 1 ,...,s d ,v }∈ G x ( { s 1 , . . . , s d , v } ) = 1 | ( V ) d | . With the random walk, we are no w able to ﬁnd the desired probability distribution, which w e will use to sample the extra-tight paths later. It is similar to Lemma 5.3 in [ 11 ], but here we analyze the probabilit y of an edge app earing in a path instead of a v ertex app earing. F urthermore, we b ound the probabilit y of an edge app earing at an end of the path, which we will need when w e glue the extra-tight paths together in to one extra-tight trail. 27 Lemma 6.4. L et 1 /n ≪ 1 /t ≪ α ≪ 1 /d ≤ 1 / 2 . L et G b e an n -vertex d -gr aph with δ ( G ) ≥ (1 − α ) n . L et Ω b e the set of al l extr a-tight p aths of or der t in G . Then ther e exists a pr ob ability distribution on Ω such that a r andomly chosen element P ∈ Ω has the fol lowing pr op erties: (i) for any given extr a-tight p ath Q , we have P [ P = Q ] = O t  n − t  ; (ii) for every e ∈ G , we have    P [ e ∈ P ] − e ∗ | G |    = O t  n − d − 1  wher e e ∗ is the numb er of e dges in an extr a-tight p ath of or der t ; (iii) for every e ∈ G , we have    P [ E e ] − 2 | G |    = O t  n − d − 1  wher e E e is the event that the d vertic es of e ar e forming one of the two ends of P . Pr o of. Let 1 /n ≪ 1 /t ≪ α ≪ 1 /d ≤ 1 / 2 and let x b e a 0 . 9 -normal fractional K d +1 -decomp osition whose existence is guaran teed b y Lemma 6.2 . Let Y = ( Y 1 , Y 2 , . . . ) b e the random walk deﬁned via x as abov e. Consisten t with [ 11 ], w e will use Pr as the probabilit y measure corresponding to the random walk whereas P will denote the desired probabilit y measure on Ω . Let B b e the even t that Y 1 , . . . , Y t are pairwise distinct. Let Q ∈ Ω b e an y path with v ertices q 1 , . . . , q t in that order. Then w e deﬁne the distribution on Ω via P [ P = Q ] : = Pr  { Y 1 = q 1 , . . . , Y t = q t } ∪ { Y 1 = q t , . . . , Y t = q 1 }|B  . W e already know that Pr[ Y 1 = q 1 , . . . , Y d = q d ] = 1 | ( V ) d | = 1 d ! | G | = O α  n − d  . F urthermore, for ev ery − → Z i = ( Y i − d +1 , . . . , Y i ) and every v ∈ V ( G ) , we hav e the following b ound on the transition using that x is 0 . 9 -normal: Pr[ Y i +1 = v | Y i − d +1 , . . . , Y i ] = x ( Z i ∪ { v } ) ≤ 0 . 9 − 1 · 1 n = O  n − 1  . Th us, the chain rule yields Pr[ Y 1 = q 1 , . . . , Y t = q t ] = O t ( n − t ) . The num b er of walks of order t which are not self-a voiding is O t  n t − 1  . Hence, Pr[ B c ] = O t  n t − 1  · O t ( n − t ) = O t  n − 1  . Finally , w e get P [ P = Q ] = Pr[ Y 1 = q 1 , . . . , Y t = q t ] + Pr[ Y 1 = q t , . . . , Y t = q 1 ] Pr[ B ] = O t  n − t  . This prov es (i) . F or (ii) and (iii) , we deﬁne the following set: E W : =  ( j 1 , . . . , j d ) ∈ [ t ] d |∃ i ∈ [ t − d + 1] : i = j 1 < j 2 < · · · < j d ≤ i + d  . The set E W con tains all ordered tuples of indices whose vertices in the walk form an edge in the extra-tight path. Fix an e ∈ G . F or each ( j 1 , . . . , j d ) ∈ E W , the n umber of walks of order t which are not self-av oiding and where { Y j 1 , . . . , Y j d } = e is O t  n t − d − 1  and thus Pr[ {{ Y j 1 , . . . , Y j d } = e } ∩ B c ] = O t  n t − d − 1  · O t ( n − t ) = O t  n − d − 1  . F urthermore, w e get Pr[ {{ Y j 1 , . . . , Y j d } = e } ∩ B |B ] = 1 Pr[ B ] (Pr[ { Y j 1 , . . . , Y j d } = e ] − Pr[ {{ Y j 1 , . . . , Y j d } = e } ∩ B c ]) = 1 1 − O t ( n − 1 )  d ! | ( V ) d | − O t  n − d − 1   = 1 | G | − O t  n − d − 1  , where we used Lemma 6.3 and Pr[ B ] = 1 − O t  n − 1  for the second equality . Note that E e = ( {{ Y 1 , . . . , Y d } = e } ∩ B ) ∪ ( {{ Y t − d +1 , . . . , Y t } = e } ∩ B ) is a disjoint union. Hence, we can immediately conclude P [ E e ] = Pr[ {{ Y 1 , . . . , Y d } = e } ∩ B |B ] + Pr[ {{ Y t − d +1 , . . . , Y j t } = e } ∩ B |B ] = 2 | G | − O t  n − d − 1  , 28 whic h sho ws (iii) . F or (ii) , we can conclude P [ e ∈ P ] = Pr   [ { j 1 ,...,j d }∈ E W  { Y j 1 , . . . , Y j d } = e  ∩ B    B   = X { j 1 ,...,j d }∈ E W Pr  { Y j 1 , . . . , Y j d } = e  ∩ B  Pr[ B ] ≥ e ∗ | G | − O t  n − d − 1  , where w e used Pr[ B ] ≤ 1 for the inequality . F or the other direction in (ii) , we use Pr[ B ] = 1 − O t  n − 1  and Pr  { Y j 1 , . . . , Y j d } = e  ∩ B  ≤ Pr  { Y j 1 , . . . , Y j d } = e  = 1 | G | (b y Lemma 6.3 ) to conclude P [ e ∈ P ] = X { j 1 ,...,j d }∈ E W Pr  { Y j 1 , . . . , Y j d } = e  ∩ B  Pr[ B ] ≤ e ∗ · 1 | G | 1 − O t ( n − 1 ) = e ∗ | G |  1 + O t  n − 1  = e ∗ | G | + O t  n − d − 1  . No w, w e are ready to pro ve the existence of a packing of the constant-length extra-tight paths co vering almost all edges. There, w e will use the following nibble theorem. Theorem 6.5 (Theorem 1.2 in [ 9 ]) . L et 1 / ∆ ≪ δ, 1 /d ≤ 1 / 2 and ε : = δ / (50 d 2 ) . L et H b e a d -gr aph with ∆ 1 ( H ) ≤ ∆ and with ∆ 2 ( H ) ≤ ∆ 1 − δ as wel l as | H | ≤ e xp  ∆ ε 2  . Supp ose that W is a set of at most exp  ∆ ε 2  weight functions ω : H → R ≥ 0 with ω ( H ) ≥ max e ∈ H ω ( e )∆ 1+ δ . Then ther e exists a matching M in H such that ω ( M ) = (1 ± ∆ − ε ) ω ( H ) / ∆ for al l ω ∈ W . Lemma 6.6. Supp ose 1 /n ≪ γ ≪ α ≪ 1 /d ≤ 1 / 2 . L et G b e an n -vertex d -gr aph with δ ( G ) ≥ (1 − α ) n . Then ther e exists a p acking P of extr a-tight p aths such that (i) the leftover L , of al l e dges of G not c over e d by any of the p aths of the p acking, satisﬁes ∆( L ) ≤ γ n ; (ii) e ach S ∈  V ( G ) d − 1  app e ars at an end of at most γ n of these p aths. Pr o of. Let t : = 1 /γ be an integer, and δ , β b e such that 1 /n ≪ δ ≪ β ≪ 1 /t ≪ α ≪ 1 /d . Set N : = n t − 1 / 2 and let P : = { P i : i ∈ [ N ] } be a set of N extra-tight paths of order t sampled indep enden tly according to the distribution of Lemma 6.4 . F or an edge e ∈ G , let X e : = { i ∈ [ N ] : e ∈ P i } . Similarly , let W e b e the set of indices i ∈ [ N ] where e app ears at an end of P i . By Item (ii) and Item (iii) of Lemma 6.4 , w e hav e E [ | X e | ] = N · e ∗ | G |  1 ± O t  n − 1  = e ∗ n t − 1 / 2 | G |  1 ± O t  n − 1  and E [ | W e | ] = N · 2 | G |  1 ± O t  n − 1  = 2 n t − 1 / 2 | G |  1 ± O t  n − 1  where, as b efore, e ∗ is the num b er of edges in an extra-tight path of order t . Therefore, the Chernoﬀ bound (Theorem 1.11 ) and a union b ound yield that, with high probability , | X e | = (1 ± β / 3) e ∗ n t − 1 / 2 | G | = Θ t  n t − d − 1 / 2  and | W e | = (1 ± β / 3)2 n t − 1 / 2 | G | = Θ t  n t − d − 1 / 2  (13) hold for all e ∈ G . Righ t now, the paths in P are not necessarily unique. In the next step, w e wan t to delete paths with the same edge set. Ho wev er, we also hav e to make sure that there is no edge e where to o many paths are deleted that con tain e . Therefore, w e ha ve to b ound ho w man y redundan t paths contain a ﬁxed edge e . Fix an edge e ∈ G . F or i < j ∈ [ N ] , let Y i,j b e the indicator v ariable of the even t { e ∈ P i = P j } and set Z e : = P i d . L et F b e a we akly r e gular d -gr aph on f vertic es and let G b e an F -divisible ( ε, ξ , f , d ) -sup er c omplex on n vertic es. Then G has an F -de c omp osition. W e will use this theorem in the pro of of the Sup ercomplex Lemma to partition G ′ ∪ L in to extra-tight cycles. Note that extra-tigh t cycles are not weakly regular, but this will not be a problem b ecause of a result from [ 13 ] (cf. Lemma 7.9 ) which allo ws us to pass from a graph which is not w eakly regular to a graph which is. Ho wev er, as mentioned in the statemen t of the Sup ercomplex Lemma, we also need that eac h of the extra-tigh t cycles has at most d vertices outside of U . T o guarantee this, we will ﬁnd a “sk ew” sup ercomplex, where eac h f -set in the sup ercomplex has at most d vertices outside of U . Deﬁnition 7.4. F or d < f , a d -graph G and U ⊆ V ( G ) , we deﬁne the skew ( G, f , d ) -c omplex G skew to b e the complex generated b y all f -sets S where G [ S ] is a clique and | U ∩ S | ≤ d . If we can ﬁnd a sup ercomplex G super ⊆ G skew and partition G super in to extra-tigh t cycles of order f , then each extra-tight cycle will ha ve at most d vertices outside of U since its vertices form an f -set in G skew . In the following, we give a pro of outline of the Supercomplex Lemma 7.1 , stating all the relev an t lemmas. Afterw ards, w e will pro ve the Supercomplex Lemma assuming all the relev ant lemmas. These will then b e prov en in the succeeding subsections. T o shorten the statements, we introduce the follo wing terminology: Deﬁnition 7.5. A d -graph is called ( D , ρ ) -rich in a set U ⊆ V ( G ) if for all A ⊆  V ( G ) d − 1  with | A | ≤ D , we ha ve   T S ∈ A G ( d ) ( S ) ∩ U   ≥ ρn . A complex G is called ( D , ρ ) -rich in a set U ⊆ V ( G ) if G ( d ) is ( D, ρ ) -ric h in U . An extra-tight cycle in a d -graph G with U ⊆ V ( G ) is valid if it has at most d v ertices outside of U . Pro of outline for the Sup ercomplex Lemma 7.1 . In a ﬁrst step, we ﬁnd a skew sup ercomplex G super in the sk ew ( G, f , d ) -complex G skew . This is given b y the following lemma where the 1.1 can b e replaced by any num b er bigger than 1: Lemma 7.6 (Existence of sk ew sup ercomplex) . L et 1 /n ≪ ε ≪ ξ , ρ ≪ η ≪ α , µ ≪ 1 /D , 1 /f ≪ 1 /d ≤ 1 / 2 with α < µ 1 . 1 . L et G b e a d -gr aph on n vertic es and U ⊆ V ( G ) with | U | = µn and δ ( G ) ≥ (1 − α ) n and G skew the skew ( G, f , d ) -c omplex. Then ther e is a sub c omplex G super ⊆ G skew that is an ( ε, ξ , f , d ) - sup er c omplex, which is ( D, ρ ) -rich in U , and wher e ∆  G ( d ) super  ≤ ηn . 33 The d -lay er G ( d ) super is exactly the d -graph G ′ whose existence the Sup ercomplex Lemma guarantees. T o prov e its absorbing prop ert y , we then assume that we hav e a small edge set L . W e cov er this edge-set with v alid extra-tight cycles using the follo wing lemma, where G ′ tak es the role of G : Lemma 7.7. L et 1 /n ≪ γ ≪ γ ′ ≪ ε, ξ , ρ ≪ 1 /D ≪ 1 /f ≪ 1 /d ≤ 1 / 2 . Supp ose G is a d -gr aph on n vertic es that is ( D , ρ ) -rich on a set U ⊆ V ( G ) . F urthermor e, let L ⊆  V ( G ) d  b e disjoint to G with ∆( L ) ≤ γ n . Then ther e is a c ol le ction F of e dge-disjoint c opies of valid E C ( d ) f such that (i) L ⊆ S F ∈F F ⊆ L ∪ G ; (ii) ∆( S F ∈F F ) ≤ γ ′ n . The previous lemma used a few edges of G ′ , but the ma jority of edges of G ′ is still not cov ered b y v alid extra-tight cycles. The goal is to co ver them no w. F or that, we hav e to make sure that it is still a reasonable sup ercomplex and that it is reasonably rich. The following lemma shows that this is the case: Lemma 7.8 ((i) is shown in [ 13 ], Lemma 5.9 (v)) . L et 1 /n ≪ γ ≪ ε, ξ ≪ 1 /D , 1 /d, 1 /f with f > d . L et G b e a c omplex on n vertic es and let S b e an d -gr aph on V ( G ) with ∆( S ) ≤ γ n . (i) If G is an ( ε, ξ , f , d ) -sup er c omplex, then G − S is an  2 ε, ξ / 2 , f , d  -sup er c omplex. (ii) If G is ( D , ρ ) -rich in U ⊆ V ( G ) , then G − S is ( D , ρ/ 2) -rich. Th us, we can still use Theorem 7.3 to partition the remaining edges of G ′ in to extra-tight cycles. Ho wev er, Theorem 7.3 still imposes t wo obstacles: It can only decomp ose a sup ercomplex into a graph F which is weakly regular (and extra-tigh t cycles are not weakly regular), and the sup ercomplex m ust b e F -divisible. The ﬁrst obstacle is easily dealt with b y the following already known result: Lemma 7.9 ([ 13 ], Lemma 9.2) . L et 2 ≤ d < f . L et F b e any d -gr aph on f vertic es. Ther e exists a we akly r e gular d -gr aph F ∗ on at most 2 f · f ! vertic es which has an F -de c omp osition. Deﬁnition 7.10. Let F ∗ f b e the weakly regular d -graph which has an E C ( d ) f -decomp osition whose exis- tence is giv en b y the previous lemma. Th us, w e can partition the sup ercomplex in to copies of F ∗ f and th us in to copies of E C ( d ) f using Theorem 7.3 as so on as it is F ∗ f -divisble. This is done with the follo wing lemma: Lemma 7.11 (Degree Fixing Lemma) . L et 1 /n ≪ γ ≪ ρ ≪ 1 /D ≪ 1 /f ≪ 1 /d ≤ 1 / 2 . L et G b e a d -gr aph on n vertic es that is ( D , ρ )-rich on a set U ⊆ V ( G ) . F urthermor e, assume that d 2 | deg G ( v ) for al l v ∈ V ( G ) \ U . Then ther e exist disjoint subsets S and H of G such that (i) S ⊆  U d  ; (ii) H has an E C ( d ) f -de c omp osition wher e e ach c opy of E C ( d ) f is valid; (iii) G − S − H is F ∗ f -divisible; (iv) ∆( S ∪ H ) ≤ γ n . With all the relev ant lemmas stated, we will now prov e the Supercomplex Lemma, following the pro of outline ab ov e. Pr o of of the Sup er c omplex L emma 7.1 . Let 1 /n ≪ γ ≪ γ ′ ≪ ε ≪ ξ , ρ ≪ η ≪ α, µ ≪ 1 /D , 1 /f ≪ 1 /d ≤ 1 / 2 with α < µ 1 . 1 . Let G skew b e the sk ew ( G, f , d ) -complex and G super ⊆ G skew the ( ε, ξ , f , d ) - sup ercomplex that is ( D , ρ ) -rich in U and where ∆  G ( d ) super  ≤ µn . W e claim that G ′ : = G ( d ) super is the d -graph whose existence is claimed by the Sup ercomplex Lemma 7.1 . Note that G ′ is ( D , ρ ) -rich and that ∆( G ′ ) ≤ µn is already given. Th us, w e just hav e to c heck the absorbing prop ert y of G ′ . F or this, let L ⊆ G b e an y edge set with ∆( L ) ≤ γ n such that d 2 | deg G ′ ∪ L ( v ) for every v ∈ V ( G ) \ U . W e hav e to ﬁnd a packing of v alid extra-tight cycles in G ′ ∪ L , eac h of order at least f , cov ering all edges in ( G ′ ∪ L ) \ ( G ′ ∪ L )[ U ] . Applying Lemma 7.7 with G 7 . 7 : = G ′ , we get a collection F of edge-disjoint copies of v alid E C ( d ) f in L ∪ G with L ⊆ S F ∈F F and ∆  S F ∈F F  ≤ γ ′ n . Let G ′′ : = G ′ − S F ∈F F = ( G ′ ∪ L ) − S F ∈F F . 34 By Lemma 7.8 , G ′′ is still ( D , ρ/ 2) -rich. F urthermore, we ha ve d 2 | deg G ′′ ( v ) for all v ∈ V ( G ) \ U since d 2 | deg G ′ ∪ L ( v ) and d 2 | deg S F ∈F F ( v ) . Th us, w e can apply Lemma 7.11 with G 7 . 11 : = G ′′ , ρ 7 . 11 : = ρ/ 2 . W e get disjoin t subsets S ⊆ G ′′ [ U ] and H ⊆ G ′′ where H has a decomposition into v alid E C ( d ) f , G ′′′ : = G ′′ − S − H is F ∗ f -divisible and ∆( S ∪ H ) ≤ γ n . Applying Lemma 7.8 with γ 7 . 8 : = 2 γ , we get that G super [ G ′′′ ] is an (2 ε, ξ / 2 , f , d ) -sup ercomplex. Since it is also F ∗ f divisible and F ∗ f is weakly regular, w e can apply Theorem 7.3 with G 7 . 3 : = G [ G ′′′ ] ε 7 . 3 : = 2 ε , ξ 7 . 3 : = ξ / 2 and get an F ∗ f -decomp osition. Since F ∗ f is E C ( d ) f -divisible, this giv es us an E C ( d ) f - decomp osition of G ′′ . By deﬁnition of G super [ G ′′′ ] ⊆ G skew , every E C ( d ) f in that decomp osition is v alid. Hence, we are done. 7.2 Pro of of Lemma 7.7 and Lemma 7.8 The only things left to prov e are the Lemm as 7.6 , 7.7 , 7.8 (ii) , and 7.11 . W e start with the tw o shortest pro ofs which do not need extra preparations. Pr o of of L emma 7.7 . W e w ant to apply Lemma 5.1 . Let L = { e 1 , . . . , e m } . Since ∆( L ) ≤ γ n , w e ha ve m ≤ γ n d . Supp ose { v 1 , . . . , v d } ∈ E C ( d ) f . F or each j ∈ [ m ] , let T j b e E C ′ : = E C ( d ) f − { v 1 , . . . , v d } and X j = { v 1 , . . . , v d } . Pic k D large enough such that E C ′ has degeneracy at most D rooted at X j . By assumption, G fulﬁlls the minim um degree in U condition of Lemma 5.1 . Finally , let Λ j b e the G -lab elling of ( T j , X j ) that sends the v ertices of { v 1 , . . . , v d } to the vertices of e j ∈ L . By ∆( L ) ≤ γ n , eac h S ⊆ V ( G ) with | S | ∈ [ d − 1] is con tained in at most γ n d −| S | edges of L and, therefore, at most γ n d −| S | man y Λ j ro ot S . Th us, w e can apply Lemma 5.1 with γ 5 . 1 : = 2 γ and get, for every j ∈ [ m ] a Λ j -faithful embedding of ( T j , X j ) that are edge-disjoin t, where ϕ j ( v ) ∈ U for all v ∈ V ( T j ) \ X j and where ∆( S j ∈ [ m ] ϕ j ( T j )) ≤ γ ′ n . Let F : = { ϕ j ( T j ) ∪ { e j } : j ∈ [ m ] } . Then F is the collection of edge-disjoint copies of v alid E C ( d ) f that fulﬁll (i) and (ii) . Pr o of of L emma 7.8 (ii) . Let A ⊆  V ( G ) d − 1  b e an arbitrary set with | A | ≤ D . Since G is ( D , ρ ) -rich, we kno w that   T e ∈ A G ( d ) ( e ) ∩ U   ≥ ρn . Each e ∈ A can be con tained in at most γ n sets of S . Thus,   T e ∈ A ( G − S ) ( d ) ( e ) ∩ U   ≥ ( ρ − D γ ) n . 7.3 Pro of of the Degree Fixing Lemma 7.11 The goal of this section is to prov e the Degree Fixing Lemma 7.11 . T o pro ve this, we will mainly use the follo wing v ersion of [ 13 , Lemma 9.4]. On one hand, our version is simpler than [ 13 , Lemma 9.4] since it only considers the case O = ∅ and G is assumed to b e F -divisible. On the other hand, w e hav e an additional condition that the F -copies that partition H hav e at most d v ertices outside of a predeﬁned vertex set U . Note that the D − D ∗ of [ 13 , Lemma 9.4] is an H here and the H ∪ D ∗ of [ 13 , Lemma 9.4] is G − H here. Lemma 7.12. L et 1 /n ≪ γ ≪ ξ , 1 /f ∗ and d ∈ [ f ∗ − 1] . L et F b e a d -gr aph. L et F ∗ b e a d -gr aph on f ∗ vertic es which has an F -de c omp osition. L et G b e an F -divisble d -gr aph on n vertic es and U ⊆ V ( G ) such that for al l A ⊆  V ( G ) d − 1  with | A | ≤  f ∗ − 1 d − 1  , we have   T S ∈ A G ( S ) ∩ U   ≥ ξ n . Then ther e exists a sub gr aph H ⊆ G such that (i) ∆( H ) ≤ γ − 1 ; (ii) G − H is F ∗ -divisible; (iii) H has an F -de c omp osition wher e e ach c opy of F has at most d vertic es outside of U . Since the pro of of this lemma is very similar to the pro of of [ 13 , Lemma 9.4], we omit it here. Broadly sp eaking, the lemma enables us to turn an E C ( d ) f -divisible sup ercomplex in to an F ∗ f -divisible sup ercomplex. Thus, w e only hav e to ﬁnd a w ay to turn G in to an E C ( d ) f -divisible sup ercomplex. By Lemma 4.6 , w e kno w that w e only need to adjust the n umber of edges and the 1-degrees of G in order 35 to make it E C ( d ) f -divisible. The follo wing lemma shows ho w this can be done b y only deleting edges inside G [ U ] . Lemma 7.13. L et 1 /n ≪ γ ≪ ρ ≪ 1 /D ≪ 1 /f , 1 /d ≤ 1 / 2 . L et G b e a d -gr aph on n vertic es that is ( D , ρ ) -rich on a set U ⊆ V ( G ) . F urthermor e, assume that d 2 | deg G ( v ) for al l v ∈ V ( G ) \ U . Then ther e is a subset S ⊆ G [ U ] with ∆( S ) ≤ γ n such that d 2 | deg G − S ( v ) holds for al l v ∈ V ( G ) and wher e ( f d ) | | G − S | . Pr o of. T ake any d 2 -regular d -graph whose num b er of edges is divisible by f d and let { v 1 , . . . , v d } b e one of its edges. Add a new vertex v 0 and replace { v 1 , . . . , v d } by { v 0 , v 2 , v 3 , . . . , v d } . Let H b e the resulting graph. In this graph, all vertices hav e a degree that is divisible b y d 2 except for v 0 whose degree is 1 and v 1 whose degree is congruen t to − 1 mo d d 2 . By embedding a cop y of H into G and putting its edges in to S , w e can decrease the degree of the image of v 0 in G − S by 1 mo d d 2 and increase the degree of the image of v 1 in G − S by 1 mo d d 2 . In this wa y , w e can adjust the degree of every v ertex in U like we did it in the pro of of Theorem 1.1 . First, we take up to f d arbitrary edges of G [ U ] in to S such that ( f d ) | | G − S | . In particular, the d -uniform Handshake Lemma implies now that the sum of v ertex degrees in G − S is divisible by d 2 . After this step, we w ant to adjust the degree b y only moving copies of H into S . In the follo wing set T , w e collect all pairs ( u, w ) where we w ant to embed a cop y of H in such a wa y that v 0 is mapp ed to u and v 1 is mapp ed to w . If w e mov e this copy of H into S , then the degree of u in G − S will decrease b y 1 (mo d d 2 ) whereas the degree of w will increase by 1 (mo d d 2 ) . Claim. There is a digraph T on the v ertex set V ( G ) such that the following conditions hold: (i) for all v ∈ V ( G ) , deg G − S ( v ) − deg + T ( v ) + deg − T ( v ) ≡ 0 (mo d d 2 ) ; (ii) for all v ∈ V ( G ) , deg + T ( v ) + deg − T ( v ) ≤ 5 d 2 . Pr o of of claim: The claim is prov ed in the same w ay as the claim in the pro of of Theorem 1.1 . − Let T be the digraph giv en by the claim with an arbitrary edge ordering. W e will apply Lemma 5.1 with m : = | T | ≤ 5 d 2 n , γ ′ 5 . 1 : = γ / 2 , γ 5 . 1 suitable small, and G 5 . 1 : = G − S . If ( u, w ) is the j -th edge of T , w e ha ve T j : = H , X j = { v 0 , v 1 } , Λ j ( v 0 ) = u , and Λ( v 1 ) = w . One can easily see that all conditions of Lemma 5.1 are satisﬁed. Th us, w e get Λ j -faithful em b eddings ϕ j of ( T j , X j ) in to U such that the images are pairwise edge- disjoin t and also edge-disjoint to the at most 3 d 2 edges in S . F urthermore, w e hav e ∆( S j ∈ [ m ] ϕ j ( T j )) ≤ γ n/ 2 . By adding the edges of all the images to S , w e get that all 1-degrees in G − S are divisible b y d 2 , ( f d ) | | G − S | , and ∆( S ) ≤ γ n . No w, w e are ready to pro ve the Degree Fixing Lemma. Here, we only need to apply the previous lemma to mak e G E C ( d ) f -divisible and then Lemma 7.12 to mak e that graph F ∗ f -divisible. Pr o of of the De gr e e Fixing L emma 7.11 . By Lemma 7.13 , there is a subset S ⊆ G [ U ] with ∆( S ) ≤ γ n suc h that for G ′ : = G − S , we hav e d 2 | deg G ′ ( v ) holds for all v ∈ V ( G ) , and where ( f d ) | | G ′ | . This, together with Lemma 4.6 , sho ws that G ′ is E C ( d ) f -divisible. By Lemma 7.8 , we get that G ′ is ( D , ρ/ 2) -rich. Th us, we can apply Lemma 7.12 with F 7 . 12 : = E C ( d ) f , F ∗ 7 . 12 : = F ∗ f , ξ 7 . 12 : = ρ/ 2 , and G 7 . 12 : = G ′ using that G ′ is ( D , ρ/ 2) -rich. W e get a subgraph H ⊆ G ′ ( d ) suc h that ∆( H ) ≤ γ − 1 , G ′ − H is F ∗ f -divisible, and H has an E C ( d ) f -decomp osition where eac h cop y of F has at most d vertices outside of U . 7.4 Pro of of the Existence of a Skew Sup ercomplex (Lemma 7.6 ) The ﬁnal step is to prov e Lemma 7.6 , i.e. the existence of a ( D , ρ ) -rich, ( ε, ξ , f , d ) -sup ercomplex G super ⊆ G skew with ∆  G ( d ) super  ≤ η n . First, w e show that we can ignore the maxim um degree condition b ecause w e get it for free in the end. This is done b y the following previously known result: Lemma 7.14 ([ 13 ], Corollary 5.19) . L et 1 /n ≪ ε, ξ ≪ η , 1 /f and d ∈ [ f − 1] . Supp ose that G is an ( ε, ξ , f , d ) -sup er c omplex on n vertic es and that H ⊆ G ( d ) is a r andom sub gr aph obtaine d by including every e dge of G ( d ) indep endently with pr ob ability η . Then with high pr ob ability, G [ H ] is a (4 ε, ξ 2 , f , d ) - sup er c omplex. 36 Corollary 7.15. L et 1 /n ≪ ε, ξ , ρ ≪ η , 1 /D, 1 /f ≤ 1 /d . Supp ose that G is an ( ε, ξ , f , d ) -sup er c omplex on n vertic es which is ( D , ρ ) -rich in U ⊆ V ( G ) . Then ther e is a sub c omplex G ′ ⊆ G that is a (4 ε, ξ 2 , f , d ) - sup er c omplex, ( D , ρ 2 ) -rich in U and wher e ∆( G ′ ( d ) ) ≤ ηn . Pr o of. Let H ⊆ G ( d ) b e a random subgraph obtained by including every edge of G ( d ) indep enden tly with probabilit y η / 2 . By Lemma 7.14 , G [ H ] is a (4 ε, ξ 2 , f , d ) -supercomplex with high probabilit y . F urthermore, eac h ( d − 1) -set is exp ected to hav e a degree of at most ( η/ 2) n . Chernoﬀ and union bound imply that ∆( G [ H ] ( d ) ) ≤ η n holds with high probability as well. Finally , for eac h A ⊆  V ( G ) d − 1  with | A | ≤ D , we ha ve   T S ∈ A G ( d ) ( S ) ∩ U   ≥ ρn . Thus, we exp ect that   T S ∈ A G [ H ] ( d ) ( S ) ∩ U   ≥ η D ρn . Using Chernoﬀ and union b ound again, w e get that G [ H ] is ( D, ρ 2 ) -ric h with high probabilit y . Th us, we just hav e to ﬁnd a ( D , ρ ) -rich, ( ε, ξ , f , d ) -supercomplex in G skew . T o do this, we ha ve to consider the precise deﬁnition of a sup ercomplex. This can also b e found in [ 13 ]. W e rep eat it here for completeness. The crucial prop erty appearing in the next deﬁnition is that of r e gularity , which means that every d -set of a given complex G is contained in roughly the same num b er of f -sets. If we view G as a complex whic h is induced by some d -graph, this means that every edge lies in roughly the same num b er of cliques of size f . Deﬁnition 7.16 ([ 13 ], Deﬁnition 4.1) . Let G b e a complex on n v ertices, f ∈ N and d ∈ [ f − 1] 0 , with 0 ≤ ε, g , ξ ≤ 1 . W e sa y that G is (i) ( ε, g , f , d ) - r e gular , if for all e ∈ G ( d ) , we hav e    G ( f ) ( e )    = ( g ± ε ) n f − d ; (ii) ( ξ , f , d ) - dense , if for all e ∈ G ( d ) , we hav e    G ( f ) ( e )    ≥ ξ n f − d ; (iii) ( ξ , f , d ) - extendable , if G ( d ) is empty or there exists a subset X ⊆ V ( G ) with | X | ≥ ξ n such that for all e ∈  X d  , there are at least ξ n f − d man y ( f − d ) -sets Q ⊆ V ( G ) \ e suc h that  Q ∪ e d  \{ e } ⊆ G ( d ) . W e say that G is a ful l ( ε, ξ , f , d ) - c omplex if G is • ( ε, g , f , d ) -regular for some g ≥ ξ , • ( ξ , f + d, d ) -dense, • ( ξ , f , d ) -extendable. W e sa y that G is an ( ε, ξ , f , d ) - c omplex if there exists an f -graph Y on V ( G ) such that G [ Y ] is a full ( ε, ξ , f , d ) -complex. Note that G [ Y ] ( d ) = G ( d ) (recall that d < f ). Deﬁnition 7.17 ([ 13 ], Deﬁnition 4.3) . Let G b e a complex. W e say that G is an ( ε, ξ , f , d ) -sup er c omplex if for every h ∈ [ d ] 0 and every set B ⊆ G ( h ) with 1 ≤ | B | ≤ 2 h , we ha ve that T b ∈ B G ( b ) is an ( ε, ξ , f − h, d − h ) -complex. Deﬁnition 7.18. Let G b e a complex and U ⊆ V ( G ) . W e say that a set e ∈ G has typ e i with resp ect to U if | e ∩ ( V ( G ) \ U ) | = i . By G i , we denote the set of sets of G with type i . W e start b y proving a lemma ab out G skew . Morally , it says that for each v ertex set A , num b er s ∈ [ | A | , d + f ] , and type j ∈ [ d ] 0 , there are roughly as man y w ays to extend A to an s -set of type j in G skew as one would exp ect. Since the deﬁnition of sup ercomplexes requires that every T b ∈ B G ( b ) has certain prop erties, w e not only sho w it for G skew , but also for ev ery suc h in tersection. Lemma 7.19. L et 1 /n ≪ α, µ ≪ 1 /f ≪ 1 /d ≤ 1 / 2 . L et G b e a d -gr aph on n vertic es and U ⊆ V ( G ) with | U | = µn and δ ( G ) ≥ (1 − α ) n and G skew the skew ( G, f , d ) -c omplex. Th en the fol lowing pr op erties ar e fulﬁl le d for al l h ∈ [ d ] 0 , B ⊆ G ( h ) skew with 1 ≤ | B | ≤ 2 h and G ′′ : = T b ∈ B G skew ( b ) , d ′ : = d − h , f ′ : = f − h , α ′ : = f µ − 1 (2 d +1 · d ) d α , n ′ : = n −   S b ∈ B b   : 37 (i) for every vertex set A ⊆ V ( G ′′ ) of size a ∈ [ d ′ , d ′ + f ′ ] and typ e k ∈ [ d ′ ] 0 , every s ∈ [ a, f ′ + d ′ ] and every j ∈ [ k , d ′ ] , ther e ar e  1 ± α ′  (1 − µ ) j − k ( j − k )! · µ s − j − a + k ( s − j − a + k )! n ′ s − a vertex sets S ⊆ V ( G ′′ ) c ontaining A of size s and typ e j such that  S d ′  \  A d ′  ⊆ G ′′ ( d ′ ) . (ii) for every e dge A ∈ G ′′ of size a ∈ [ d ′ , d ′ + f ′ ] and typ e k ∈ [ d ′ ] 0 , every s ∈ [ a, f ′ + d ′ ] and every j ∈ [ k , d ′ ] , ther e ar e  1 ± α ′  (1 − µ ) j − k ( j − k )! · µ s − j − a + k ( s − j − a + k )! n ′ s − a vertex sets S ⊆ V ( G ′′ ) c ontaining A of size s and typ e j such that S ∈ G ′′ ( s ) . Pr o of. Fix an h ∈ [ d ] 0 , B ⊆ G ( h ) skew with 1 ≤ | B | ≤ 2 h and deﬁne G ′′ , d ′ , f ′ , α ′ , n ′ as in the statement of the lemma. T o chec k (i) , take a set A ⊆ V ( G ′′ ) of size a ∈ [ d ′ , d ′ + f ′ ] and t yp e k ∈ [ d ′ ] 0 . F urthermore, ﬁx an s ∈ [ a, f ′ + d ′ ] and a j ∈ [ k, d ′ ] . Clearly , there are at most  (1 − µ ) n ′  j − k ( j − k )! · ( µn ′ ) s − j − a + k ( s − j − a + k )! v ertex sets S ⊆ V ( G ′′ ) that con tain A and are of size s and t yp e j . Hence, w e only hav e to low er b ound the num b er of v ertex sets S ⊆ V ( G ′′ ) containing A of size s and type j such that  S d ′  \  A d ′  ⊆ G ′′ ( d ′ ) . W e choose the vertices of S one at a time. Supp ose w e ha ve already pic ked m v ertices (including those of A ). W e hav e to pick the next vertex v in the follo wing wa y: F or every b ∈ B and every ( d − h − 1) - subset T of the already chosen v ertices, the set b ∪ T ∪ { v } must b e in G ( d ) skew . By the minimum degree condition of G , we get that eac h choice of b and T can exclude at most αn v ertices. Hence and b ecause of   S b ∈ B b   ≤ 2 h · h , there are deﬁnitely at least  1 − µ −  m +2 h · h d − 1  α  n ≥  1 − µ − (2 d +1 · d ) d α  n ′ v ertices outside of U which we can pic k next. Similarly , there are at least  µ − (2 d +1 · d ) d α  n ′ v ertices inside of U . Therefore, the n umber of vertex sets S with the desired prop erties is at least   1 − µ − (2 d +1 · d ) d α  n ′  j − k ( j − k )! ·   µ − (2 d +1 · d ) d α  n ′  s − j − a + k ( s − j − a + k )! ≥  1 − µ − 1 (2 d +1 · d ) d α  (1 − µ )  j − k ( j − k )! ·  1 − µ − 1 (2 d +1 · d ) d α  µ  s − j − a + k ( s − j − a + k )! n ′ s − a ≥  1 − ( s − a ) µ − 1 (2 d +1 · d ) d α  (1 − µ ) j − k ( j − k )! · µ s − j − a + k ( s − j − a + k )! n ′ s − a In the last step, w e used Bernoulli’s inequalit y . Therefore, the n umber is (1 ± α ′ ) (1 − µ ) j − k ( j − k )! · µ s − j − a + k ( s − j − a + k )! n ′ s − a . Condition (ii) is c heck ed similarly: Let A ∈ G ′′ b e an edge of size a ∈ [ d ′ , d ′ + f ′ ] and type k ∈ [ d ′ ] 0 and ﬁx an s ∈ [ a, f ′ + d ′ ] and a j ∈ [ k, d ′ ] . There are at most  (1 − µ ) n ′  j − k ( j − k )! · ( µn ′ ) s − j − a + k ( s − j − a + k )! edges S ∈ G ′′ that con tain A and are of size s and type j . Hence, we only hav e to low er b ound the n umber of edges S ∈ G ′′ con taining A of size s and type j . W e choose the v ertices of S one at a time. Suppose we ha ve already pick ed m vertices (including those of A ) such that these m vertices form an edge S ′ in G ′′ . Then w e wan t to pick the next vertex v in suc h a w ay that S ′′ : = S ′ ∪ { v } is also an edge in G ′′ . This is the case if for ev ery b ∈ B , S ′′ ∪ b is 38 in G skew . By the deﬁnition of G skew , this is the case if  S ′′ ∪ b d  ⊆ G ( d ) skew . Since S ′ is in G , ev ery d -subset of S ′′ ∪ b that does not contain v is already known to b e in G ( d ) skew . W e hav e to choose v suc h that for every T ∈  S ′ ∪ b d − 1  , T ∪ { v } ∈ G ( d ) skew . By the minimum degree condition of G skew , eac h choice of T can exclude at most αn vertices. Hence and because of   S b ∈ B b   ≤ 2 h · h , there are deﬁnitely at least  1 − µ −  m +2 h · h d − 1  α  n ≥  1 − µ −  2 d +1 · d  d α  n ′ v ertices outside of U which w e can pick next. Similarly , there are at least  µ −  2 d +1 · d  d α  n ′ v ertices inside of U . The remaining calculations are exactly as in the pro of of (i) . In the following deﬁnition, w e will list all the properties w e need to show that a complex is an ( ε, ξ , f , d ) -complex for certain ε and ξ . Deﬁnition 7.20. W e say that a complex G on n vertices is ( α, µ, f , d, c, C ) -advantage ous if the following prop erties are fulﬁlled with w j : =  µ 3 f d  j for all j ∈ [ d ] 0 : • for all i ∈ [ d ] 0 , e ∈ G ( d ) i and ℓ ∈ [ i, d ]    G ( f + d ) ℓ ( e )    ≥ c · µ f − ℓ + ℓ ( f + d − 1 d − 1 ) · n f (15) • for all i ∈ [ d ] 0 , Q ∈ G ( f ) i and ℓ ∈ [ i, d ]    G ( f + d ) ℓ ( Q )    ≤ C · µ f − ℓ + i − i · ( f − 1 d − 1 ) + ℓ · ( f + d − 1 d − 1 ) · n d (16) • for all i ∈ [ d ] 0 , e ∈ G ( d ) i and ℓ ∈ [ i, d ]    G ( f ) ℓ ( e )    =  1 ± α  (1 − µ ) ℓ − i ( ℓ − i )! · µ f − ℓ − d + i ( f − ℓ − d + i )! · w − 1 i d Y t =0 w ( ℓ t )( f − ℓ d − t ) t · n f − d (17) • for all i ∈ [ d ] 0 , e ∈  V ( G ) d  of type i , and ℓ ∈ [ i, d ] , the num b er of vertex sets S of type ℓ and size f con taining e such that  S d  \{ e } ⊆ G ( d ) is at least c · µ f − ℓ − d + ℓ · ( f − 1 d − 1 ) · n f − d . (18) The G ′ in the follo wing lemma will b e our ﬁnal ( ε, ξ , f , d ) -sup ercomplex. Lemma 7.21. L et 1 /n ≪ ρ ≪ α, µ ≪ c, 1 /C ≪ 1 /D , 1 /f ≪ 1 /d ≤ 1 / 2 with α < µ 1 . 1 . L et G b e a d -gr aph on n vertic es and U ⊆ V ( G ) with | U | = µn and δ ( G ) ≥ (1 − α ) n . Then ther e is a sub c omplex G ′ ⊆ G skew that is ( D, ρ ) -rich in U and such that for al l h ∈ [ d ] 0 , B ⊆ G ′ ( h ) with 1 ≤ | B | ≤ 2 h , G ′′ : = T b ∈ B G ′ ( b ) is (2 α ′ , µ, f ′ , d ′ , c, C ) -advantage ous on n ′ vertic es wher e d ′ : = d − h , f ′ : = f − h , α ′ : = f µ − 1 (2 d +1 · d ) d α , n ′ : = n −   S b ∈ B b   . Pr o of. W e create a random set X of d -edges in the following wa y . F or all i ∈ [ d ] 0 , any edge in G ( d ) i is tak en into X with probability w i : =  µ 3 f d  i indep enden tly of eac h other. Let G ′ : = G skew [ X ] . W e will sho w no w that with high probability , G ′ will fulﬁll all desired prop erties. First, w e sho w ( D , ρ ) -richness in U . F or this, let A ⊆  V ( G ′ ) d − 1  with | A | ≤ D . W e need to sho w   T S ∈ A G ′ ( d ) ( S ) ∩ U   ≥ ρn . Indeed, for eac h e ∈ A , there are at most αn elements in U such that e ∪ { u } ∈ G b y the minim um degree condition of G . Therefore,   T S ∈ A G ( d ) ( S ) ∩ U   ≥ ( µ − | A | α ) n . Thus, the exp ected size of T S ∈ A G ′ ( d ) ( S ) ∩ U is at least w | A | d ( µ − | A | α ) n ≥  µ 3 f d  dD ( µ − D α ) n. By Chernoﬀ ’s b ound (Theorem 1.11 ) and a union b ound o ver the polynomially many choices of A , w e get that with high probabilit y , G ′ is ( D , ρ ) -rich in U . 39 Next, let h ∈ [ d ] 0 , B ⊆ G ′ ( h ) with 1 ≤ | B | ≤ 2 h and G ′′ : = T b ∈ B G ′ ( b ) , d ′ : = d − h , f ′ : = f − h , α ′ : = f µ − 1 (2 d +1 · d ) d α , n ′ : = n −   S b ∈ B b   . W e will show that G ′′ is (2 α ′ , µ, f ′ , d ′ , c, C ) -adv an tageous. Note that 1 /n ′ ≪ α ′ , µ ≪ 1 /f ≪ 1 /d holds b ecause α < µ 1 . 1 . Let A ⊆ V ( G ′′ ) b e a vertex set of size a ∈ [ d ′ , d ′ + f ′ ] and type k ∈ [ d ′ ] 0 . Let s ∈ [ a, f ′ + d ′ ] and j ∈ [ k , d ′ ] . W e deﬁne E s,j ( A ) to b e the n umber of vertex sets S ⊆ V ( G ′′ ) of size s and type j containing A such that  S d ′  \  A d ′  ⊆ G ′′ ( d ′ ) . By Lemma 7.19 (i) , we get E [ E s,j ( A )] =  1 ± α ′  (1 − µ ) j − k ( j − k )! · µ s − j − a + k ( s − j − a + k )! · d ′ Y t =0 w − ( k t )( a − k d ′ − t ) t d ′ Y t =0 w ( j t )( s − j d ′ − t ) t n ′ s − a . Similarly , if A is even an edge in T b ∈ B G skew ( b ) , we can deﬁne E ′ s,j ( A ) to b e the num b er of edges S ∈ G ′′ ( s ) of type j con taining A . Then w e get with the help of prop erty Lemma 7.19 (ii) E  E ′ s,j ( A ) | A ∈ G ′′  =  1 ± α ′  (1 − µ ) j − k ( j − k )! · µ s − j − a + k ( s − j − a + k )! · d ′ Y t =0 w − ( k t )( a − k d ′ − t ) t d ′ Y t =0 w ( j t )( s − j d ′ − t ) t n ′ s − a . Therefore and by Chernoﬀ (Theorem 1.11 ), with high probability , the following hold for all h ∈ [ d ] 0 , B ⊆ G ′ ( h ) with 1 ≤ | B | ≤ 2 h and G ′′ : = T b ∈ B G ′ ( b ) simultaneously 2 : a = d ′ k = i s = f ′ + d ′ j = ℓ    G ′′ ( f ′ + d ′ ) ℓ ( e )    =  1 ± 2 α ′  (1 − µ ) ℓ − i ( ℓ − i )! · µ f ′ − ℓ + i ( f ′ − ℓ + i )! · w − 1 i d ′ Y t =0 w ( ℓ t )( f ′ + d ′ − ℓ d ′ − t ) t · n ′ f ′ = Θ d ′ ,f ′  µ f ′ − ℓ + i · µ − i · µ P d ′ t =0 t ( ℓ t )( f + d ′ − ℓ d ′ − t ) · n ′ f ′  = Θ d ′ ,f ′  µ f ′ − ℓ + ℓ ( f ′ + d ′ − 1 d ′ − 1 ) · n ′ f ′  for all i ∈ [ d ′ ] 0 , e ∈ G ′′ ( d ′ ) i and ℓ ∈ [ i, d ′ ] ; a = f ′ k = i s = f ′ + d ′ j = ℓ    G ′′ ( f ′ + d ′ ) ℓ ( Q )    =  1 ± 2 α ′  (1 − µ ) ℓ − i ( ℓ − i )! · µ f ′ − ℓ + i ( d ′ − ℓ + i )! d ′ Y t =0 w − ( i t )( f − i d ′ − t ) t d ′ Y t =0 w ( ℓ t )( f ′ + d ′ − ℓ d ′ − t ) t n ′ d ′ = Θ d ′ ,f ′  µ f ′ − ℓ + i · µ − P d ′ t =0 t ( i t )( f ′ − i d ′ − t ) · µ P d ′ t =0 t · ( ℓ t )( f ′ + d ′ − ℓ d ′ − t ) · n ′ d ′  = Θ d ′ ,f ′  µ f ′ − ℓ + i − i · ( f ′ − 1 d ′ − 1 ) + ℓ · ( f ′ + d ′ − 1 d ′ − 1 ) · n ′ d ′  for all i ∈ [ d ′ ] 0 , Q ∈ G ′′ ( f ′ ) i and ℓ ∈ [ i, d ′ ] ; a = d ′ k = i s = f ′ j = ℓ    G ′′ ( f ′ ) ℓ ( e )    =  1 ± 2 α ′  (1 − µ ) ℓ − i ( ℓ − i )! · µ f ′ − ℓ − d ′ + i ( f ′ − ℓ − d ′ + i )! · w − 1 i d ′ Y t =0 w ( ℓ t )( f ′ − ℓ d ′ − t ) t · n ′ f ′ − d ′ for all i ∈ [ d ′ ] 0 , e ∈ G ′′ ( d ′ ) i and ℓ ∈ [ i, d ′ ] ; a = d ′ k = i s = f ′ j = ℓ | E f ′ ,ℓ ( e ) | =  1 ± 2 α ′  (1 − µ ) ℓ − i ( ℓ − i )! · µ f ′ − ℓ − d ′ + i ( f ′ − ℓ − d ′ + i )! · w − 1 i d ′ Y t =0 w ( ℓ t )( f ′ − ℓ d ′ − t ) t · n ′ f ′ − d ′ = Θ d ′ ,f ′  µ f ′ − ℓ − d ′ + i · µ − i · µ P d ′ t =0 t ( ℓ t )( f ′ − ℓ d ′ − t ) · n ′ f ′ − d ′  = Θ d ′ ,f ′  µ f ′ − ℓ − d ′ + ℓ · ( f ′ − 1 d ′ − 1 ) · n ′ f ′ − d ′  2 In the following equations, we frequently use the identit y P d j =0 j ·  i j  f − i d − j  = i ·  f − 1 d − 1  which holds since b oth count the num b er of wa ys to pick a committee with d members out of i w omen and f − i men and then choose a woman from the committee as presiden t. 40 for all i ∈ [ d ′ ] 0 , e ∈  V ( G ′′ ) d ′  of type i , and ℓ ∈ [ i, d ′ ] . All that is left to do no w, is to show that the fact that all the T b ∈ B G ′ ( b ) are adv antageous implies that they all are ( ε, ξ , f , d ) -complexes for certain ε and ξ . In that pro of, w e use the following tw o results: Prop osition 7.22 ([ 13 ]) . L et f > d ≥ 2 and let e and J b e disjoint sets with | e | = d and | J | = f . L et G b e the c omplete c omplex on e ∪ J . Ther e exists a function ψ : G ( f ) → R such that (i) for al l e ′ ∈ G ( d ) , P Q ∈ G ( f ) ( e ′ ) ψ ( Q ∪ e ′ ) = ( 1 , e ′ = e 0 , e ′  = e ; (ii) for al l Q ∈ G ( f ) , | ψ ( Q ) | ≤ 2 d − j ( d − j )! ( f − d + j j ) , wher e j : = | e ∩ Q | . Lemma 7.23. Consider the upp er-triangular n × n -matrix A : =      a 11 a 12 . . . a 1 n 0 a 22 . . . a 2 n . . . . . . . . . . . . 0 0 . . . a nn      with non-ne gative entries, p ositive diagonal entries and such that a ij a j j ≤ 1 2 n for al l i < j . Then, A      x 1 x 2 . . . x n      =      1 1 . . . 1      has a solution with 1 2 a ii ≤ x i ≤ 1 a ii for al l i ∈ [ n ] . Pr o of. W e prov e the statement b y induction from i = n down to 1. F or x n , this holds by the last equation. Supp ose, it has already been sho wn for x n , . . . , x i +1 . Then, the i -th equation yields x i = 1 − P n k = i +1 a ik x k a ii . By assumption and induction h yp othesis, a ik x k is non-negative, whence x i ≤ 1 a ii . F urthermore, w e get x i = 1 − P n k = i +1 a ik x k a ii ≥ 1 − P n k = i +1 a ik a kk a ii ≥ 1 − n − i 2 n a ii ≥ 1 2 a ii . W e are now ready to show that the G ′ from Lemma 7.21 is indeed an ( ε, ξ , f , d ) -sup ercomplex. Lemma 7.24. L et 1 /n ≪ ε ≪ ξ ≪ α, µ ≪ c, 1 /C ≪ 1 /f ≪ 1 /d ≤ 1 / 2 . L et G b e a c omplex on n vertic es which is ( α, µ, f , d, c, C ) -advantage ous. Then G is an ( ε, ξ , f , d ) -c omplex. Pr o of. Recall that w j =  µ 3 f d  j for all j ∈ [ d ] 0 b y the deﬁnition of ( α, µ, f , d, c, C ) -adv an tageous. Consider the ( d + 1) × ( d + 1) -matrix A = ( a iℓ ) i,ℓ ∈ [ d ] 0 with a iℓ : = ( (1 − µ ) ℓ − i ( ℓ − i )! · µ f − ℓ − d + i ( f − ℓ − d + i )! · w − 1 i Q d j =0 w ( ℓ j )( f − ℓ d − j ) j if i ≤ ℓ 0 if ℓ < i. This matrix is upp er-triangular with non-negative en tries. F or i ∈ [ d ] 0 , we get a ii = µ f − d ( f − d )! · w − 1 i d Y j =0 w ( i j )( f − i d − j ) j = µ f − d ( f − d )! ·  3 f d µ  i ·  µ 3 f d  P d j =0 j · ( i j )( f − i d − j ) = µ f − d ( f − d )! ·  3 f d µ  i ·  µ 3 f d  i · ( f − 1 d − 1 ) . 41 W e can conclude a ii = Θ f ,d  µ f − d − i + i · ( f − 1 d − 1 )  . (19) F urthermore, a iℓ a ℓℓ = (1 − µ ) ℓ − i ( ℓ − i )! · µ i − ℓ · ( f − d )! ( f − d + i − ℓ )! · w ℓ w i ≤ µ i − ℓ · f ℓ − i · w ℓ w i = (3 d ) i − ℓ ≤ 1 2( d + 1) for all i < ℓ . By Lemma 7.23 , the linear equations A      g ′ 0 g ′ 1 . . . g ′ d      =      1 1 . . . 1      ha ve a solution with 1 2 a ii ≤ g ′ i ≤ 1 a ii for all i ∈ [ d ] 0 . By ( 19 ), g ′ i = Θ f ,d  µ − f + d + i − i · ( f − 1 d − 1 )  . Setting g : = 1 /g ′ d and g i : = g g ′ i , we get A      g 0 g 1 . . . g d      =      g g . . . g      (20) with g d = 1 and g = Θ f ,d  µ f − 2 d + d · ( f − 1 d − 1 )  (21) g i = Θ f ,d  µ − d + i +( d − i ) ( f − 1 d − 1 )  (22) for all i ∈ [ d − 1] 0 . W e w an t to ﬁnd f -subsets Y such that G [ Y ] is a full ( ε, ξ , f , d ) -complex. Assume that there is a function ψ : G ( f ) → [0 , 1] such that for every e ∈ G ( d ) , we hav e X Q ′ ∈ G ( f ) ( e ) ψ ( Q ′ ∪ e ) = ( g / 2) n f − d and g ℓ / 4 ≤ ψ ( Q ) ≤ 1 for all Q ∈ G ( f ) ℓ . W e can then take every Q ∈ G ( f ) indep enden tly into Y with probabilit y ψ ( Q ) . W e will show now that in that case, G [ Y ] is a full ( ε, ξ , f , d ) -complex. Indeed, we hav e E    G [ Y ] ( f ) ( e )    = ( g / 2) n f − d . Thus, Theorem 1.11 implies P h    G [ Y ] ( f ) ( e )     =  1 ± n − ( f − d ) / 2 . 01  ( g / 2) n f − d i ≤ 2 exp − n − 2 2 . 01 ( f − d ) 3 ( g / 2) n f − d ! ≤ e − n 0 . 004 This shows that with high probabilit y , G [ Y ] is  n − ( f − d ) / 2 . 01 , g / 2 , f , d  -regular and, hence, ( ε, g / 2 , f , d ) - regular. Note that g / 2 > ξ b y ( 21 ). F or the density , let e ∈ G ( d ) and Q ∈ G ( f + d ) ( e ) . Then P h Q ∈ G [ Y ] ( f + d ) ( e ) i = Y Q ′ ∈ ( Q ∪ e f ) ψ ( Q ′ ) ≥ ( g 0 / 4) ( f + d f ) ( 22 ) ≥ µ d ( f − 1 d − 1 )( f + d d ) . Since   G ( f + d ) ( e )   ≥    G ( f + d ) d ( e )    ( 15 ) ≥ µ f + d · ( f + d − 1 d − 1 ) · n f , E h G [ Y ] ( f + d ) ( e ) i ≥ µ f + d ( f − 1 d − 1 )( f + d d ) + d · ( f + d − 1 d − 1 ) n f ≥ µ 1 . 5 d ( f − 1 d − 1 )( f + d d ) n f . 42 Here, we wan t to apply McDiarmid’s Inequalit y (Theorem 1.13 ). F or each Q ∈ G ( f ) , let X Q b e the indicator v ariable that Q is in Y . If | Q ∩ e | = ℓ , then changing the v alue of X Q c hanges   G [ Y ] ( f + d ) ( e )   b y at most n ℓ . Since there are at most n f − ℓ man y suc h f -sets Q , McDiarmid’s Inequality implies P h   G [ Y ] ( f + d ) ( e )   ≤ µ 2 d ( f − 1 d − 1 )( f + d d ) n f i ≤ 2 exp  − E    G [ Y ] ( f + d ) ( e )    2 / P d ℓ =0 n 2 ℓ · n f − ℓ  ≤ exp  − n f − d − 0 . 01  . Therefore, with high probability , G [ Y ] is ( ξ , f + d, d ) -dense. Finally , we hav e to chec k extendability . Fix any set X ⊆ V ( G ) of size ξ n . Let e ∈  X d  . By ( 18 ), the n umber of ( f − d ) -sets Q ⊆ V ( G ) \ e such that  Q ∪ e d  \{ e } ⊆ G ( d ) ⊆ G [ Y ] ( d ) is at least µ f + d · ( f − 1 d − 1 ) · n f − d ≥ ξ n f − d . Therefore G [ Y ] is ( ξ , f , d ) -extendable whic h concludes the proof that G is an ( ε, ξ , f , d ) -complex. It remains to sho w that such a ψ exists. By Prop osition 7.22 , for every e ∈ G ( d ) and J ∈ G ( f + d ) ( e ) , there exists a function ψ e,J : G ( f ) → R such that (i) ψ e,J ( Q ) = 0 for all Q ⊆ e ∪ J ; (ii) for all e ′ ∈ G ( d ) , P Q ′ ∈ G ( f ) ( e ′ ) ψ e,J ( Q ′ ∪ e ′ ) = ( 1 , e ′ = e 0 , e ′  = e ; (iii) for all Q ∈ G ( f ) , | ψ e,J ( Q ) | ≤ 2 d − j ( d − j )! ( f − d + j j ) , where j : = | e ∩ Q | . W e wan t to deﬁne a function ψ : G ( f ) → [0 , 1] such that for each e ∈ G ( d ) , we get X Q ′ ∈ G ( f ) ( e ) ψ ( Q ′ ∪ e ) = ( g / 2) n f − d . Let ψ ′ : G ( f ) → [0 , 1] b e the function whic h maps an f -set of type i to g i / 2 . F or every e ∈ G ( d ) i , we deﬁne c e : = ( g / 2) n f − d − P d j = i ( g j / 2)    G ( f ) j ( e )       G ( f + d ) i ( e )    . Note that by ( 17 ) and ( 20 ), w e hav e P d j = i g j    G ( f ) j ( e )    = (1 ± α ) P d j = i g j a ij n f − d =  1 ± α  g n f − d whence | c e | ≤ αg n f − d 2    G ( f + d ) i ( e )    ( 15 ) = O c  αg µ − f + i − i ( f + d − 1 d − 1 ) · n − d  . Deﬁne ψ i : G ( f ) → [0 , 1] as ψ i : = X e ∈ G ( d ) i c e X J ∈ G ( f + d ) i ( e ) ψ e,J for every i ∈ [ d ] 0 . Note that by (i) , ψ i ( Q ) = 0 for all f -sets Q of t yp e larger than i . Finally let ψ : = ψ ′ + d X i =0 ψ i . F or every e ∈ G ( d ) i , we hav e X Q ′ ∈ G ( f ) ( e ) ψ ( Q ′ ∪ e ) = d X j = i ( g j ) / 2    G ( f ) j ( e )    + d X i ′ =0 X e ′ ∈ G ( d ) i ′ c e ′ X J ∈ G ( f + d ) i ′ ( e ′ ) X Q ′ ∈ G ( f ) ( e ) ψ e ′ ,J ( Q ′ ∪ e ) (ii) = d X j = i ( g j ) / 2    G ( f ) j ( e )    + c e    G ( f + d ) i ( e )    = ( g / 2) n f − d , 43 as desired. The last thing to c heck is that ψ ( Q ) is b et ween g ℓ / 4 and 1 for all Q ∈ G ( f ) ℓ . Let Q ∈ G ( f ) ℓ . Then | ψ ( Q ) − g ℓ / 2 | =      d X i = ℓ ψ i ( Q )      (i) ≤ d X i = ℓ X e ∈ G ( d ) i ,J ∈ G ( f + d ) i ( e ) Q ⊆ e ∪ J | c e || ψ e,J ( Q ) | (iii) ≤ d X i = ℓ    G ( f + d ) i ( Q )    · O c  αg µ − f + i − i ( f + d − 1 d − 1 ) · n − d  ( 16 ) ≤ d X i = ℓ O C  µ f − i + ℓ − ℓ · ( f − 1 d − 1 ) + i · ( f + d − 1 d − 1 ) · n d  · O c  αg µ − f + i − i ( f + d − 1 d − 1 ) · n − d  = d X i = ℓ O c,C  αg µ ℓ − ℓ · ( f − 1 d − 1 )  ( 21 ) = O c,C  α · µ f − 2 d + d · ( f − 1 d − 1 ) + ℓ − ℓ · ( f − 1 d − 1 )  = O c,C  α · µ f − d · µ − d + ℓ +( d − ℓ ) · ( f − 1 d − 1 )  ( 22 ) = O c,C  α · µ f − d · g ℓ  If α and µ are small enough, this is smaller than g ℓ / 4 . Pr o of of L emma 7.6 . Let G ′ ⊆ G skew b e the subcomplex given b y Lemma 7.21 . W e claim that G ′ is an ( ε, ξ , f , d ) -supercomplex. Let h ∈ [ d ] 0 , B ⊆ G ′ ( h ) with 1 ≤ | B | ≤ 2 h and G ′′ : = T b ∈ B G ′ ( b ) . W e ha ve to sho w that G ′′ is an ( ε, ξ , f − h, d − h ) -complex. By Lemma 7.21 , w e know that G ′′ is ( α ′ , µ, f ′ , d ′ , c, C ) - adv antageous where d ′ : = d − h , f ′ : = f − h , α ′ : = f µ − 1 (2 d +1 · d ) d α , n ′ : = n −   S b ∈ B b   . By Lemma 7.24 , this implies that G ′′ is an ( ε, ξ , f ′ , d ′ ) -complex. Thus, G ′ is an ( ε, ξ , f , d ) -sup ercomplex which is ( D , ρ ) -rich b y Lemma 7.21 . Applying Corollary 7.15 on G ′ yields the sparse sup ercomplex w e are lo oking for. 8 Concluding Remarks Reﬁned absorption. In the pro of of Theorem 1.8 one could p otentially replace the use of the iterativ e absorption method in Section 7 with “reﬁned” absorb ers, dev elop ed recently by Delcourt and P ostle [ 8 ]. V ery roughly sp eaking, they pro ve the existence of an absorber A together with a collection of cliques suc h that for any divisible leftov er L , one can decomp ose A ∪ L using only cliques from the sp eciﬁed col- lection. Crucially , this collection of cliques can b e chosen so sparse that ev ery edge lies only in a constan t n umber of them. Unfortunately for our application one w ould need an appropriate reﬁned absorption statemen t with extr a-tight cycles instead of cliques. While this sort of result quite p ossibly holds, it is not curren tly av ailable in the literature. F urthermore, even with such an alternative approach, the key nov el con tributions of our pap er (the reduction of the diameter problem to a hypergraph decomposition result via turns, the construction of the switc hers, and the approximate decomp osition lemma with bo osted parameters) would still b e necessary . Inc hing to wards geometry . It is w ell kno wn (cf. [ 22 ]) that for the purp oses of the maximum diameter of a d -dimensional p olytop e with n facets it is suﬃcient to consider those which are simple , that is every v ertex is con tained in exactly d of the facets. Each vertex of a simple p olytop e P can then be enco ded as a d -subset of its n facets, giving rise to a d -graph F = F ( P ) on vertex set [ n ] . The dual graph G ( F ) is exactly the vertex/edge graph of P . Indeed, tw o v ertices of P are connected by an edge of P if and only if the tw o corresponding d -sets hav e an intersection of size d − 1 (corresp onding to the d − 1 facets of P con taining the edge). Since the vertex/edge graph of P is connected, so is G ( F ) , and hence H s ( n, d − 1) is an upp er b ound on the diameter of p olytopes. The simplicial d -complexes ⟨F ( P ) ⟩ coming from p olytop es ha ve in fact a muc h richer topological structure, b eyond the connectivit y of their dual graph. As a ﬁrst step, the diameter of pseudomanifolds without b oundary w ere considered b y Criado and San tos [ 6 ]. The simplicial d -complex generated by a ( d + 1) -graph F is called a pseudomanifold without b oundary if every mem b er of ∂ F is con tained in exactly t wo mem b ers of F . After improv ements of [ 5 ], Bohman and Newman [ 3 ] determined the correct asymptotics of the maximum diameter H pm ( n, d ) of d -dimensional pseudomanifolds without boundary on n vertices. W e can determine the precise v alue for 44 d = 2 and every large enough n . A precise b ound for d ≥ 3 and ev ery large enough n seems to b e quite a challenge. In another direction, the nature of our construction requires the low er b ound n 0 on the num b er of v ertices to b e signiﬁcan tly large in terms of d . F or the connection of the Polynomial Hirsc h Conjecture to the Simplex Metho d, the particular range when n is linear in d is of quite a relev ance, as many of the w orst case examples are in there. Santos’ pro duct construction [ 22 ] do es giv e an exp onential low er b ound in this range, but more precise, p ossibly exact results would b e of great in terest. The optimal minim um degree condition for the existence of extra-tigh t tours. In relation to Theorem 1.7 it w ould b e v ery interesting to determine, or at least estimate, the suprem um of those α > 0 for whic h every d -graph on n > n 0 ( d, α ) vertices with δ ( G ) > (1 − α ) n , satisfying that every vertex has degree divisible b y d 2 admits an extra-tigh t Euler tour. The answer is not even known for d = 2 . A c kno wledgemen ts Stefan Glock is funded b y the Deutsc he F orsc hungsgemeinsc haft (DF G, German Researc h F oundation) – 542321564. Silas Rathke is funded b y the Deutsc he F orsc hungsgemeinsc haft (DFG, German Researc h F oundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MA TH+ (EX C-2046/1, pro ject ID: 390685689). 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The maximum diameter of $d$-dimensional simplicial complexes

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