Towards Pósa's Conjecture for $3$-graphs

T o w ards P´ osa’s Conjecture for 3-graphs Debmaly a Bandy opadhy a y ∗ Allan Lo † Ric hard Mycroft ‡ 31st Marc h 2026 Abstract W e pro v e that ev ery 3-graph H on n vertices with minimum codegree δ 2 ( H ) ≥ 7 n/ 9 + o ( n ) con tains the square of a tigh t Hamilton cycle. This strengthens a theorem of Bedenknech t and Reiher that δ 2 ( H ) ≥ 4 n/ 5 + o ( n ) is sufficien t. The central nov elty of our argumen ts is an improv ed understanding of the connectivit y structure of 3-graphs with large minimum co degree. 1 In tro duction A fundamen tal question in graph theory is to determine optimal conditions on a host graph G — often expressed in terms of the minimum degree δ ( G ) — whic h guarantee the existence of a given structure in G . Dirac’s theorem that ev ery graph G on n ≥ 3 v ertices with δ ( G ) ≥ n 2 admits a Hamilton cycle is an arc hetypal result of this kind. Since then, a v ast b o dy of theory has b een dev elop ed describing sufficien t conditions for em b eddings in graphs, hypergraphs, directed graphs and other combinatorial settings. One of the most influen tial problems in dev eloping this theory w as the P´ osa–Seymour conjecture stating that every graph G on n ≥ 3 vertices with δ ( G ) ≥ r r +1 n con tains the r -th p o wer C r of a Hamilton cycle, that is, the graph obtained from a Hamilton cycle b y adding the edges of a copy of K r +1 on each set of r + 1 consecutiv e vertices of the cycle (P´ osa’s conjecture w as the case r = 2 concerning squares of Hamilton cycles, and Seymour [ 26 ] later proposed the more general statement). The difficulty of this conjecture arises principally due to the high level of connectivit y of p o wers of cycles. Indeed, a p erfect K r +1 -tiling (a spanning collection of vertex-disjoin t copies of K r +1 ) can b e viewed as a less-connected analogue of C r , since lo cally each clique has the same structure as r + 1 consecutive v ertices of C r . The Ha jnal–Szemer ´ edi theorem [ 9 ], which can b e prov ed by relativ ely simple techniques, states that the same minim um degree condition as in the P´ osa–Seymour conjecture forces G to contain a p erfect K r +1 -tiling. By contrast, after a great deal of attention and partial results b y man y researchers [ 6 , 7 , 15 , 16 ], it w as only by using extensive nov el machinery including Szemer´ edi’s Re gularit y lemma and the Blo w-up lemma that the P´ osa–Seymour conjecture w as finally pro ven for sufficien tly large n by Koml´ os, Sark¨ ozy and Szemer´ edi [ 17 ]. F or small v alues of n the conjecture remains op en to this day . The question we consider here is the generalisation of the P´ osa–Seymour conjecture to 3-graphs (i.e. 3-uniform h yp ergraphs — see Section 2 for this definition and for other standard terms pertaining to hypergraphs). Sp ecifically , we consider minimum co degree conditions whic h ensure the existence of the square of a tight Hamilton cycle in a 3-graph. The minimum c o de gr e e of a 3-graph H , denoted b y δ 2 ( H ), is the largest m for which every pair of vertices of H is con tained in at least m edges. A 3-graph contains the squar e of a tight Hamilton cycle if its vertices can b e cyclically ordered so ∗ Sc ho ol of Mathematics, Univ ersity of Birmingham, E-mail : d.bandyopadhyay@bham.ac.uk † Sc ho ol of Mathematics, Univ ersity of Birmingham, E-mail : s.a.lo@bham.ac.uk . ‡ Sc ho ol of Mathematics, Univ ersity of Birmingham, E-mail : r.mycroft@bham.ac.uk . 1 that every set of four consecutiv e v ertices forms a tetrahedron (that is, a copy of K 3 4 , the complete 3-graph on four v ertices). Our main result is that a minimum co degree of 7 n/ 9 + o ( n ) suffices for this purp ose. Theorem 1.1. F or al l α > 0 ther e exists n 0 ∈ N such that every 3 -gr aph H on n ≥ n 0 vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n c ontains the squar e of a tight Hamilton cycle. This strengthens a previous result of Bedenknech t and Reiher [ 4 ], who show ed that a minimum co degree of 4 n/ 5 + o ( n ) is sufficient. P av ez-Sign ´ e, Sanhueza-Matamala and Stein [ 21 ] subsequently generalised their argument to give a b ound for the r -th p o w er of a tigh t Hamilton cycle in a k -graph for each r and k , with the same 4 n/ 5 + o ( n ) b ound for the square of a tight Hamilton cycle in a 3-graph. As in the graph case, the cen tral difficult y in pro ving Theorem 1.1 arises due to the high level of connectedness of the structure w e are aiming to embed. Indeed, the b est-possible minimum co degree condition for a p erfect tetrahedron tiling in a 3-graph was identified as essen tially 3 n/ 4 b y Lo and Markstr¨ om [ 19 ] and b y Keev ash and Mycroft [ 14 ] (the latter authors in fact gav e the precise optimal b ound for sufficien tly large n ). Ev ery four consecutiv e v ertices in the square of a tigh t Hamilton cycle induce a tetrahedron, so the perfect tetrahedron tiling can be seen as a less-connected analogue of the structure we seek to em b ed. More widely , our understanding of em b eddings of spanning structures in hypergraphs is p o or in comparison to the well-dev elop ed theory of spanning structures in graphs, and a key reason for this is our p oor understanding of how to handle connectivity in hypergraphs. Our pro of of Theorem 1.1 makes a significant con tribution on this sp ecific p oint: the key ideas which enable us to impro ve on the work of Bedenknec ht and Reiher [ 4 ] is a b etter understanding of the connectivit y structure of 3-graphs satisfying the giv en minimum co degree condition, as outlined in Section 1.1 . W e also recommend the surveys b y R¨ odl and Ruci ´ nski [ 23 ] and by Zhao [ 27 ] for further discussion of Hamilton cycles and tilings in uniform h yp ergraphs. W e do not b eliev e that the b ound of Theorem 1.1 is optimal. Indeed, w e conjecture that a condition of 3 n/ 4 + o ( n ) is likely to b e sufficient for the existence of the square of a tight Hamilton cycle in a graph on n vertices. Conjecture 1.2. F or al l α > 0 ther e exists n 0 ∈ N such that every 3 -gr aph H on n ≥ n 0 vertic es with δ 2 ( H ) ≥ (3 / 4 + α ) n c ontains the squar e of a tight Hamilton cycle. A slight mo dification of a construction b y Pikhurk o [ 22 ] shows Conjecture 1.2 , if true, would b e b est-possible up to the αn error term. This is the following prop osition, whic h w e pro ve in Section 2.1 . Prop osition 1.3. F or e ach n > 4 ther e exists a 3 -gr aph H on n vertic es with δ 2 ( H ) = ⌊ 3 n/ 4 ⌋ − 2 that do es not c ontain the squar e of a tight Hamilton cycle. The minim um degree condition of Theorem 1.1 is crucial to our pro of argumen ts, and significant new ideas would b e needed to reduce the gap in minim um co degree b et ween Theorem 1.1 and Prop osition 1.3 . W e consider this to b e a highly significan t op en problem in extremal graph theory , since its resolution app ears to require a muc h b etter understanding of ho w to handle connectivity in uniform h yp ergraphs under minimum codegree conditions, which is lik ely to pro ve useful for man y other problems in this area. 1.1 Key ideas and definitions The cen tral new ideas in our w ork relate to connectivit y prop erties in 3-graphs. Sp ecifically , we study the tigh t connectivity of the tetr ahe dr al gr aph T ( H ) of a 3-graph H of large minimum co degree. This is defined to b e the 4-graph with vertex set V ( H ) whose edges are all sets of four vertices whic h induce a copy of K 3 4 in H . 2 Let H b e a k -graph. F or edges e, f ∈ E ( H ), a tight walk from e to f in H is a sequence of (not necessarily disjoint) v ertices W = ( v 1 , v 2 , . . . , v ℓ ) with e = { v 1 , . . . , v k } and f = { v ℓ − k +1 , . . . , v ℓ } such that { v i , . . . , v i + k − 1 } ∈ E ( H ) for each i ∈ [ ℓ − k + 1]. In other w ords, W b egins with the vertices of e in some order, concludes with the vertices of f in some order, and each set of k consecutive vertices in W forms an edge of H . A set E ⊆ E ( H ) of edges of H is tightly c onne cte d if for all e, f ∈ E there exists a tigh t w alk from e to f . Observe that the relation on E ( H ) in whic h e ∼ f if there exists a tigh t walk from e to f is an equiv alence relation; the equiv alence classes of this relation are the tight c omp onents of H . W e say that H is tightly connected if E ( H ) is tightly c onne cte d , meaning that H has a single tight comp onen t. It is straightforw ard to show that if H is a 3-graph on n vertices with δ 2 ( H ) > 4 n/ 5, then T ( H ) is tightly connected. Indeed, note that any tw o copies of K 3 4 in H sharing three vertices are tightly connected in T ( H ). Since H is tightly connected (c.f. [ 24 , Lemma 2.1]), it suffices to show that giv en any t wo edges { x, y , z } and { w , y , z } in H , there exists a vertex u suc h that b oth { x, y , z , u } and { w , y , z , u } form a copy of K 3 4 in H . Since δ 2 ( H ) > 4 n/ 5, we ma y do this b y choosing u to b e a common neighbour of the fiv e pairs xy , xz , y z , w y and w z . A v arian t of this argumen t play ed a key role in Bedenknech t and Reiher’s pro of [ 4 ]. A k ey step in the pro of of Theorem 1.1 is to show that the w eaker minimum co degree condition δ 2 ( H ) > 7 n/ 9 is also sufficien t to ensure that T ( H ) is tigh tly connected. This is the follo wing lemma, whose pro of is the sub ject of Section 3 . Lemma 1.4. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 . Then T ( H ) is tightly c onne cte d. A squar e d-tight-walk in a 3-graph H is an ordered sequence W = ( v 1 , v 2 , . . . , v ℓ ) of vertices of H in which every set of four consecutive v ertices forms a copy of K 3 4 in H . So squared-tigh t-walks in H are precisely tigh t walks in the tetrahedral graph T ( H ). W e refer to the ordered triples ( v 1 , v 2 , v 3 ) and ( v ℓ − 2 , v ℓ − 1 , v ℓ ) as the initial triple and the final triple of W resp ectiv ely , and we refer to b oth triples as ends of W . F or conv enience of notation w e often write ordered triples as, for example, v = ( v 1 , v 2 , v 3 ). W e say that W is fr om u to v if W has initial triple u and final triple v . Of particular imp ortance are squar e d-tight-p aths in H , which are squared-tigh t-walks in H in which all v ertices in the sequence are distinct. Since the minimum degree condition of Lemma 1.4 ensures that ev ery edge of H extends to a cop y of K 3 4 in H , the conclusion of Lemma 1.4 can b e equiv alently reform ulated by saying that for all e, f ∈ E ( H ) there is a squared-tight-w alk from e to f . How ev er, this squared-tigh t-walk is not necessarily a squared-tight-path in H . By applying Lemma 1.4 to the reduced 3-graph of H obtained from an application of the Regular Slice Lemma of Allen, B¨ ottcher, Co oley and Mycroft [ 1 ], we can sho w that indeed there must b e a squared-tigh t-path from e to f in H for all e, f ∈ E ( H ). In fact we give a stronger result – the following connecting lemma – whic h allo ws us to find squared- tigh t-paths linking each of several pairs of edges. Moreov er, w e may insist that these squared-tight- paths are pairwise v ertex-disjoint and a void a giv en small set X of ‘forbidden’ vertices. F or a set P = { P 1 , . . . , P ℓ } of squared-tight-paths, we write V ( P ) for S 1 ≤ i ≤ ℓ V ( P i ). The asymptotic notation a ≪ b used here — which is standard in this field — is defined formally in Section 2 . Lemma 1.5 (Connecting Lemma) . L et 1 /n ≪ ψ ≪ α . L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n . L et X ⊆ V ( H ) b e such that | X | ≤ ψ n . L et x 1 , y 1 , . . . , x s , y s b e vertex-disjoint e dges in H \ X with s ≤ ψ n . Then ther e exists a set P = { P 1 , . . . , P s } of vertex-disjoint squar e d-tight- p aths in H \ X such that for e ach i ≤ s , P i is fr om x i to y i , | V ( P ) | ≤ ψ 1 / 2 n and | V ( P ) ∩ X | = ∅ . The deriv ation of Lemma 1.5 from Lemma 1.4 is presented in Section 5 , and the definitions and results concerning hypergraph regularity used in this argumen t are presented in Section 4 . The remaining elements of the pro of of Theorem 1.1 follow an absorbing strategy , an approach whic h is no w fairly commonplace for extremal problems in com binatorics. This com bines the afore- men tioned connecting lemma with the following t wo lemmas; w e note that Lemma 1.5 plays a part 3 in the pro of of each of these results, and that the pro of of Lemma 1.7 also makes use of hypergraph regularit y . Lemma 1.6 (Absorption Lemma) . L et 1 /n ≪ β ≪ α . L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n . Then ther e exists a squar e d-tight-p ath P in H on at most 7 √ β n vertic es such that for al l sets L ⊆ V ( H ) \ V ( P ) with | L | ≤ β 2 n , ther e is a squar e d-tight-p ath P ′ with V ( P ′ ) = V ( P ) ∪ L such that P and P ′ have the same initial and final triples. Lemma 1.7 (P ath Co v er Lemma) . L et 1 /n ≪ γ ≪ α . L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n . Then for al l disjoint p airs of or der e d triples of vertic es e 1 and e 2 that ar e e dges of H , ther e is a squar e d-tight-p ath fr om e 1 to e 2 in H which c overs al l but at most γ n vertic es of H . W e give the pro of of Lemma 1.6 in Section 6 and the pro of of Lemma 1.7 in Section 7 . Our main result, Theorem 1.1 , then follo ws by com bining Lemmas 1.5 , 1.6 and 1.7 . Pr o of of The or em 1.1 . Let 1 /n ≪ γ ≪ β ≪ α . By Lemma 1.6 , there is a squared-tigh t-path P in H on at most 7 √ β n vertices with the absorbing property stated in Lemma 1.6 . Let e 1 and e 2 b e the initial and final triples of P resp ectiv ely , so e 1 and e 2 are disjoint edges of H . Let int( P ) = V ( P ) \ ( V ( e 1 ) ∪ V ( e 2 )). By Lemma 1.7 with H \ in t( P ) pla ying the role of H , there is a squared- tigh t-path ˆ P from e 2 to e 1 in H which co vers all v ertices of H \ int( P ) except for a set L of at most γ n vertices. As γ < β 2 , by Lemma 1.6 there is a squared-tigh t-path P ′ from e 1 to e 2 in H suc h that V ( P ′ ) = V ( P ) ∪ L . Then P ′ ∪ ˆ P is the square of a tight Hamilton cycle in H . 2 Preliminary results and notation W e will use constant hierarc hies in our statements as follo ws: the phrase “ a ≪ b ” means “for ev ery b > 0 there exists a 0 > 0 such that for all 0 < a ≤ a 0 the following statements hold”. Hierarc hies with more constants are defined analogously , and we implicitly assume that if a term of the form 1 /m app ears in such a hierarch y then m is a p ositive in teger. W e omit flo ors and ceilings when they do not affect the calculations. F or n ∈ N w e write [ n ] := { 1 , . . . , n } , and for a set V and k ∈ N we write  V k  to denote the set of all subsets of V of size k . F or notational simplicit y , we write v 1 . . . v k for the set { v 1 , . . . , v k } or the ordered k -tuple ( v 1 , . . . , v k ); it will b e clear from the context whic h interpretation is intended. A k -uniform hyp er gr aph H (or a k -gr aph for brevity) consists of a vertex set V ( H ) and an edge set E ( H ) ⊆  V ( H ) k  . W e make the follo wing definitions for k -graphs H . F or a vertex subset S ⊆ V ( H ), the neighb ourho o d of S is N H ( S ) := { T ⊆ V ( H ) \ S : S ∪ T ∈ E ( H ) } and the de gr e e of S is deg H ( S ) := | N H ( S ) | . The minimum ℓ -de gr e e of H is δ ℓ ( H ) = min { deg H ( S ) : S ∈  V ( H ) ℓ  } . W e also refer to the minimum ( k − 1)-degree δ k − 1 ( H ) as the minimum c o de gr e e of H . Sometimes w e wish to consider the minimum degree ov er just those ( k − 1)-sets whic h are con tained in at least one edge of H , for which w e define the minimum p ositive c o de gr e e of H to b e δ + k − 1 ( H ) := min  deg H ( S ) : S ∈  V k − 1  , deg H ( S ) > 0  . F or a set of v ertices U ⊆ V ( H ) w e write H [ U ] to denote the subgraph induced by U , which has v ertex set U and edge set { e ∈ E ( H ) : e ⊆ U } . F or k -graphs G and H we write H \ G to denote the subgraph obtained b y deleting V ( G ) from H and H − G to denote the subgraph obtained b y deleting E ( G ) from E ( H ). F or a vertex set U ⊆ V ( H ) we define H \ U := H [ V ( H ) \ U ]. F or v ertex sets S and W , we write N H ( S, W ) = N H ( S ) ∩  W k −| S |  and deg H ( S, W ) = | N H ( S, W ) | . F or a v ertex v ∈ V ( H ), let L ( v ) denote the link gr aph of v in H , which is a ( k − 1)-graph with vertex set V ( H ) \ { v } and E ( L ( v )) = N H ( v ). F or an edge e ∈ E ( H ) we write ∂ e :=  e k − 1  , and w e write ∂ H to denote the ( k − 1)-graph on V ( H ) with edge set S e ∈ E ( H ) ∂ e . 4 No w let H b e a 3-graph on vertex set V , and let W 1 = x 1 . . . x ℓ and W 2 = y 1 . . . y m b e tigh t walks in H . If x ℓ − 1 = y 1 and x ℓ = y 2 , then we define W 1 W 2 := x 1 . . . x ℓ y 3 y 4 . . . y m . Observ e that W 1 W 2 is a tigh t walk in H . F urthermore, if W 1 and W 2 are tigh t paths in H and V ( W 1 ) ∩ V ( W 2 ) \ { y 1 , y 2 } = ∅ then W 1 W 2 is also a tight path in H . Similarly , if W 1 and W 2 are tigh t paths in H and w e also ha ve y m − 1 = x 1 and y m = x 2 and V ( W 1 ) ∩ V ( W 2 ) \ { x 1 , x 2 , y 1 , y 2 } = ∅ , then W 1 W 2 is a tigh t cycle in H . F or k , t ∈ N w e write K k t for the complete k -graph on t vertices, whic h has vertex set [ t ] and whose edges are all sets e ∈  [ t ] k  . 2.1 Extremal construction The following slight mo dification of a construction giv en by Pikhurk o [ 22 ] demonstrates that, if true, Conjecture 1.2 would b e b est-p ossible up to the αn error term. Pr o of of Pr op osition 1.3 . Let V 1 , V 2 , V 3 and V 4 b e pairwise-disjoint sets of vertices with P i ∈ [4] | V i | = n and ⌊ n/ 4 ⌋ ≤ | V i | ≤ ⌈ n/ 4 ⌉ for all i ∈ [4]. Let H b e the 3-graph with vertex set V := S i ∈ [4] V i whose edges are all triples e ∈  V 3  whic h satisfy one of the following m utually exclusive prop erties. (i) | e ∩ V 1 | = 2, (ii) | e ∩ V 1 | = | e ∩ V i | = | e ∩ V j | = 1 for distinct i, j ∈ [4] \ { 1 } , (iii) | e ∩ V i | = 3 for some i ∈ [4] \ { 1 } , or (iv) | e ∩ V i | = 1 and | e ∩ V j | = 2 for distinct i, j ∈ [4] \ { 1 } . F or xy ∈  V 2  with x ∈ V i and y ∈ V j , we then ha ve N ( xy ) =      V \ ( V 1 ∪ { x, y } ) if i = j , V \ ( V j ∪ { x } ) if i = 1  = j , V \ ( V k ∪ { x, y } ) if { i, j, k } = [4] \ { 1 } . Hence δ 2 ( H ) = ⌊ 3 n/ 4 ⌋ − 2. Ev ery cop y K of K 3 4 in H in tersects V 1 in exactly 0 or 2 vertices. Indeed, there are no edges of H in V 1 , so | K ∩ V 1 | ≤ 2. If | K ∩ V 1 | = 1, then e := V ( K ) \ V 1 is an edge of H , so our choice of E ( H ) implies that w e cannot hav e | e ∩ V i | = 1 for eac h i ∈ [4] \ { 1 } . W e must therefore ha ve | e ∩ V i | ≥ 2 for some i ∈ [4] \ { 1 } , and so K contains an edge of H with tw o vertices in V i and one in V 1 , but again this is not p ossible by our c hoice of edges. Let C = v 1 v 2 . . . v ℓ b e the square of a tight cycle in H . Note that eac h set of four consecutive v ertices of C forms a cop y of K 3 4 , and so has either 0 or 2 vertices in V 1 . It follows that either every four consecutiv e vertices of C ha ve no vertices in V 1 , or ev ery four consecutive vertices of C ha ve precisely tw o v ertices in V 1 . In the former case w e ha ve ℓ ≤ | V 2 ∪ V 3 ∪ V 4 | ≤ ⌈ 3 n/ 4 ⌉ , and in the latter case we hav e ℓ ≤ 2 | V 1 | ≤ 2 ⌈ n/ 4 ⌉ . This pro ves the prop osition. 2.2 T ur´ an density Giv en a k -graph F and n ∈ N , the T ur´ an numb er ex( n, F ) is the maximum n umber of edges that an n -vertex k -graph can ha ve without ha ving F as a subgraph. The asymptotics of this quantit y are more interesting, hence the T ur´ an density π ( F ) of F is defined as π ( F ) = lim n →∞ ex( n, F )  n k  . It is not hard to see that this limit exists for every k -graph F . Sev eral v arian ts of T ur´ an density hav e b een considered. Mubayi and Zhao [ 20 ] in tro duced the c o de gr e e T ur´ an numb er ex co ( n, F ) as the maximum d such that there exists an n -v ertex k -graph with 5 minim um co degree d which do es not ha ve F as a subgraph. They defined the c o de gr e e T ur´ an density of F as γ ( F ) = lim n →∞ ex co ( n, F ) n and prov ed that this limit exists for every k -graph F . Giv en a k -graph F and a p ositiv e integer t , we write F ( t ) to denote the t -blowup of F , which is the k -graph formed as follows. F or eac h vertex v ∈ V ( F ) let U v b e a set of t vertices, with the sets U v b eing pairwise vertex-disjoin t. The v ertex set of F ( t ) is V := S v ∈ V ( F ) U v , and a set e ∈  V k  is an edge of F ( t ) if and only if there exists an edge f ∈ F with | e ∩ U v | = 1 for each v ∈ f . W e use the follo wing result on sup ersaturation and the T ur´ an densit y and co degree T ur´ an densit y of blo wups. Sup ersaturation for T ur´ an density was disco vered by Erd˝ os and Simono vits [ 5 ], while for co degree T ur´ an density it was shown by Mubayi and Zhao [ 20 ]. The following statement can b e deduced by combining results from Keev ash’s survey [ 13 ] with those of Mubayi and Zhao [ 20 ]. Theorem 2.1 (cf. [ 13 , Lemmas 2.1, 2.2], [ 20 , Prop osition 1.4]) . L et 1 /n ≪ β ≪ α , 1 /k , 1 /m, 1 /t . L et F b e a k -gr aph on m vertic es and G b e a k -gr aph on n vertic es with | E ( G ) | ≥ ( π ( F ) + α )  n k  or δ k − 1 ( G ) ≥ ( γ ( F ) + α ) n . Then G c ontains at le ast β  n tm  c opies of F ( t ) . W e also use the follo wing result by Balogh, Clemen and Lidic k ´ y [ 3 ] which pro vides an upp er b ound on the co degree T ur´ an densit y of K 3 5 . Theorem 2.2 ([ 3 , Theorem 1.2, T able 1]) . γ ( K 3 5 ) ≤ 0 . 74 . 2.3 Probabilistic to ols Our arguments make use of the following standard probabilistic inequalities. F or a p ositiv e in teger n and p ∈ [0 , 1], the binomial distribution Bin( n, p ) is the n umber of successes from n indep enden t trials, where the probability of success in each trial is p . Now let N , n, m b e positive in tegers with n, m ≤ N , and fix a set T of size N and a subset S ⊆ T of size | S | = m . The hyp er ge ometric distribution Hyp( N , n, m ) is the distribution of the random v ariable X = | S ∩ R | , where R ⊆ T of size | R | = n is selected uniformly at random among all subsets of T with n elements. Lemma 2.3 (Marko v’s inequalit y cf. [ 12 , Equation 1.3]) . If t > 0 and X is a non-ne gative r andom variable, then P [ X ≥ t ] ≤ E X /t . Lemma 2.4 (Chernoff ’s inequalit y cf. [ 12 , Remark 2.5]) . If 0 < ε ≤ 3 / 2 and X ∼ Bin( n, p ) , then P ( | X − E X | ≥ ε E X ) ≤ 2 e − ε 2 3 E X . Lemma 2.5 (Ho effding’s inequality cf. [ 12 , Theorem 2.10]) . If 0 < ε ≤ 3 / 2 and X ∼ Hyp( N , n, m ) , then P [ | X − E X | ≥ ε E X ] ≤ 2 e − ε 2 3 E X . 3 Tigh t Connectivity In this section we prov e Lemma 1.4 , whic h states that if H is a 3-graph on n vertices with δ 2 ( H ) > 7 n/ 9 then the tetrahedral 4-graph T ( H ) of H is tightly connected. Let T be the set of distinct tight comp onen ts of T ( H ). Recall that these comp onen ts T ∈ T partition the set of copies of K 3 4 in H . Throughout this section the 3-graphs H we work with will ha ve the prop ert y that every edge of H is contained in at least one copy of K 3 4 . (Observ e that having minim um p ositiv e co degree δ + 2 ( H ) > 2 n/ 3 is a sufficient condition for this.) Moreo ver, recalling that the tight components T ∈ T partition the set of copies of K 3 4 in H , we observe that all of the copies of K 3 4 whic h con tain a given edge e ∈ E ( H ) must b e contained in the same tight comp onen t T ∈ T . 6 So for each edge e ∈ E ( H ) there is precisely one tight comp onen t T ∈ T which con tains the copies of K 3 4 con taining e (of which there is at least one); we denote this tight comp onen t by ϕ ( e ), giving an edge-colouring ϕ : E ( H ) → T of H (where the colours are the tight comp onen ts in T ). W e iden tify eac h tigh t comp onen t T ∈ T with the 3-graph on v ertex set V ( H ) and whose edges are all edges e ∈ E ( H ) with ϕ ( e ) = T ; in this wa y w e may discuss N T ( S ), N ∂ T ( S ), deg T ( S ) and deg ∂ T ( S ), as defined earlier. F or each v ∈ V ( H ), we write ϕ v ( uw ) := ϕ ( uv w ). So ϕ v is an edge-colouring of L ( v ) (the link graph of v ). Also, for eac h S ⊆ V ( H ) with | S | ≤ 2, let ϕ ( S ) = { ϕ ( e ) : e ∈ E ( H ) and S ⊆ e } , so ϕ ( S ) is the set of colours of edges whic h contain S as a subset (this set may contain multiple elements). If T ∈ ϕ ( S ), then w e say that S is in tight c omp onent T . Since t wo copies of K 3 4 whic h share three vertices must lie in the same tigh t comp onent, we ha ve the following fact. F act 3.1. L et H b e a 3 -gr aph. Supp ose that e, e ′ ∈ E ( H ) have | e ∩ e ′ | = 2 and T P ∈ ∂ e ∪ ∂ e ′ N H ( P )  = ∅ . Then e and e ′ ar e in the same tight c omp onent of T ( H ) . Our definition of ϕ ( S ) ensures that ϕ ( uv ) ⊆ ϕ ( u ) ∩ ϕ ( v ) and ϕ ( uv w ) ∈ ϕ ( uv ) ∩ ϕ ( uw ) ∩ ϕ ( v w ). The follo wing statement follows immediately , since if δ 2 ( H ) > 0 then for each pair u, v of distinct v ertices of H there is an edge of H containing b oth u and v . F act 3.2. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 0 in which e ach e dge is c ontaine d in a c opy of K 3 4 . Then for al l u, v ∈ V ( H ) we have ϕ ( u ) ∩ ϕ ( v )  = ∅ . In p articular, if ther e exists u ∈ V ( H ) satisfying | ϕ ( u ) | = 1 , then ϕ ( u ) ⊆ ϕ ( v ) for al l v ∈ V ( H ) . The next fact gives a lo wer b ound on the num b er of vertices cov ered b y each tigh t comp onent, and holds b ecause for each pair a, b of distinct v ertices of an edge e there are at most n − δ + 2 ( H ) v ertices u ∈ V ( H ) for which abu is not an edge of H , and so at least n − 3( n − δ + 2 ( H )) v ertices of H form an edge with all three pairs of vertices of e . Recall that δ + 2 ( H ) is the minimum p ositiv e co degree of H , and (for the second part of the statemen t) that all the copies of K 3 4 extending a giv en edge lie in the same T ∈ T . F act 3.3. L et α > 0 and H b e a 3 -gr aph on n vertic es with δ + 2 ( H ) ≥ (7 / 9 + α ) n . Then every e dge e ∈ E ( H ) has deg T ( H ) ( e ) ≥ (1 / 3 + 3 α ) n . If additional ly δ 2 ( H ) > 0 , then for al l xy ∈  V ( H ) 2  and T ∈ ϕ ( xy ) we have | V ( T ) | ≥ deg T ( xy ) ≥ (1 / 3 + 3 α ) n . Our next prop osition asserts that if T ( H ) has at least tw o tight comp onents, then some pair of v ertices of H is in at least t wo of these tight comp onen ts. Prop osition 3.4. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 for which T ( H ) has at le ast two tight c omp onents. Then ther e exists xy ∈  V ( H ) 2  such that | ϕ ( xy ) | ≥ 2 . Mor e over, for al l xy ∈  V ( H ) 2  , al l T 1 , T 2 ∈ ϕ ( xy ) and al l z ∈ N T 1 ( xy ) , ther e exists w z ∈ V ( H ) \ { x, y , z } with w z xy , w z y z ∈ E ( H ) and ϕ ( w z xy ) = T 2 . Pr o of. By F act 3.2 there must exist x ∈ V ( H ) with | ϕ ( x ) | ≥ 2, as otherwise by c ho osing vertices u 1 and u 2 with ϕ ( u 1 )  = ϕ ( u 2 ) we would hav e ϕ ( u 1 ) ∩ ϕ ( u 2 ) = ∅ . Fix such an x and distinct tigh t comp onents T 1 , T 2 ∈ ϕ ( x ); w e may then choose v 1 , v 2 ∈ V ( H ) \ { x } with T 1 ∈ ϕ ( xv 1 ) and T 2 ∈ ϕ ( xv 2 ). If | ϕ ( xv i ) | ≥ 2 for some i ∈ [2], then we may take y = v i ; otherwise, we may c ho ose y ∈ N ( xv 1 ) ∩ N ( xv 2 ). F or the second part of the statemen t, let xy ∈  V ( H ) 2  ha ve T 1 , T 2 ∈ ϕ ( xy ) and c ho ose z ∈ N T 1 ( xy ). By F act 3.3 w e hav e deg T 2 ( xy ) > n/ 3, so we ma y choose w z ∈ N H ( y z ) ∩ N T 2 ( xy ). Our next prop osition shows that in an y collection of nine pairs of vertices of H there are some eigh t pairs which share a common neighbour. Similarly , given a nominated pair of vertices of H and a further sev en pairs, w e can find seven pairs including the nominated pair which share a common neigh b our. F or notational simplicity , for a set P ⊆  V ( H ) 2  and P ∈ P , we write P \ P to mean P \ { P } . 7 Prop osition 3.5. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 . (i) L et P ⊆  V ( H ) 2  have |P | ≤ 9 . Then some P ′ ∈ P has T P ∈P \ P ′ N H ( P )  = ∅ . (ii) L et P ⊆  V ( H ) 2  have |P | ≤ 8 and fix P ∈ P . Then some P ′ ∈ P \ P has T P ′′ ∈P \ P ′ N H ( P ′′ )  = ∅ . Pr o of. F or (i) , note that P P ∈P deg H ( P ) > |P | (7 n/ 9) ≥ ( |P | − 2) n . Since H has n v ertices, by a veraging some x ∈ V ( H ) must b e con tained in N H ( P ) for at least |P | − 1 of the pairs P ∈ P . F or (ii) , fix W ⊆ N H ( P ) with | W | = 7 n/ 9, and observe that X P ′′ ∈P \ P | N H ( P ′′ ) ∩ W | > ( |P | − 1)(5 n/ 9) ≥ ( |P | − 3)7 n/ 9 = ( |P | − 3) | W | . By av eraging we deduce that some v ertex x ∈ W is contained in N H ( P ′′ ) ∩ W for at least |P | − 2 of the pairs P ′′ ∈ P \ P . Our pro of of Lemma 1.4 pro ceeds through three key steps. First w e show that every vertex of H is in at most t wo tigh t comp onen ts of T ( H ) in Section 3.1 . W e then sho w that T ( H ) has at most tw o tigh t comp onen ts in Section 3.2 . Finally , w e sho w that in fact T ( H ) has only one tigh t comp onen t in Section 3.3 . 3.1 Ev ery v ertex is in at most tw o tigh t comp onen ts The aim of this section is to pro ve Lemma 3.7 , which states that every vertex of H is in at most tw o tigh t comp onen ts of T ( H ). W e pro ceed by studying the prop erties of the edge-coloured link graph of a v ertex in the following lemma, leading to the conclusion that each pair of distinct vertices is in at most tw o tight comp onen ts. Lemma 3.6. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 . Then for al l v ∈ V ( H ) the fol lowing statements hold for the e dge c olouring ϕ v of L ( v ) . (i) Ther e is no pr op erly c olour e d c opy of C 4 in L ( v ) . (ii) Ther e is no c opy of C 4 in L ( v ) with 3 c olours. (iii) Ther e is no c opy of P 4 in L ( v ) with 3 c olours. (iv) F or e ach u ∈ V ( H ) \ { v } we have | ϕ ( uv ) | ≤ 2 . Pr o of. F or (i) and (ii) , observ e that if a cop y of C 4 in L ( v ) is properly coloured or has three colours, then we may write its vertices as u 0 u 1 u 2 u 3 in such a w ay that ϕ v ( u i − 1 u i )  = ϕ v ( u i u i +1 ) for each i ∈ [3]. Let P = { v u i : i ∈ [4] } ∪ { u i u i +1 : i ∈ [4] } , with addition in the index p erformed mo dulo 4, so |P | = 8. By Prop osition 3.5(ii) with v u 2 pla ying the role of P , we deduce that there exists a pair P ′ ∈ P \ { v u 2 } and a vertex x suc h that x ∈ T P ∈P \ P ′ N H ( P ). It follows that for some i ∈ [3] we ha ve x ∈ \ P ∈ ∂ v u i − 1 u i ∪ ∂ v u i u i +1 N H ( P ) , so ϕ v ( u i − 1 u i ) = ϕ v ( u i u i +1 ) by F act 3.1 , giving a con tradiction. F or (iii) , suppose for a contradiction that w xy z is a path on 4 vertices in L ( v ) with 3 colours. By F act 3.3 we hav e deg ϕ ( v yz ) ( v y ) > n/ 3. So we ma y c ho ose a v ertex u ∈ N ϕ ( v yz ) ( v y ) ∩ N H ( v w ), and then uw xy forms a cop y of C 4 in L ( v ) with at least 3 colours, contradicting (ii) . F or (iv) , supp ose for a contradiction that some v ∈ V ( H ) \ { v } has | ϕ ( uv ) | ≥ 3. Cho ose distinct tigh t comp onen ts T 1 , T 2 , T 3 ∈ ϕ ( uv ) and v ertices v 1 , v 2 and v 3 with uv v i ∈ T i for each i ∈ [3]. Let P = { uv , uv i , v v i : i ∈ [3] } , so |P | = 7. By Prop osition 3.5(ii) with uv playin g the role of P , w e obtain P ′ ∈ P \ { uv } for which T P ∈P \ P ′ N H ( P )  = ∅ . It follows that there exist distinct i, j ∈ [3] suc h that T P ∈ ∂ uv v i ∪ ∂ uv v j N H ( P )  = ∅ . By F act 3.1 w e then ha ve T i = ϕ ( uv v i ) = ϕ ( uv v j ) = T j , a con tradiction. 8 Lemma 3.7. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 . Then every vertex is in at most two tight c omp onents of T ( H ) , that is, | ϕ ( v ) | ≤ 2 for al l v ∈ V ( H ) . Pr o of. Supp ose for a contradiction that a vertex v ∈ V ( H ) is in three tight comp onen ts T 1 , T 2 , T 3 of T ( H ). F or eac h i ∈ [3] w e ha ve deg ∂ T i ( v ) > n/ 3 b y F act 3.3 , so we m ust ha v e N ∂ T i ( v ) ∩ N ∂ T j ( v )  = ∅ for some distinct i, j ∈ [3]. Assume without loss of generality that N ∂ T 1 ( v ) ∩ N ∂ T 2 ( v )  = ∅ and fix u ∈ N ∂ T 1 ( v ) ∩ N ∂ T 2 ( v ) and w ∈ N H ( uv ) ∩ N ∂ T 3 ( v ). By Lemma 3.6(iv) w e hav e ϕ v ( uw ) = ϕ ( uv w ) ∈ ϕ ( uv ) = { T 1 , T 2 } ; assume without loss of gener- alit y that ϕ v ( uw ) = T 1 . Let u ′ ∈ N T 2 ( uv ) and w ′ ∈ N T 3 ( v w ) \ { u ′ } . Then u ′ uw w ′ is a cop y of P 4 in L ( v ) with ϕ v ( u ′ u ) = T 2 , ϕ v ( uw ) = T 1 and ϕ v ( w w ′ ) = T 3 , a contradiction to Lemma 3.6(iii) . 3.2 There are at most t w o tigh t comp onen ts The aim of this section is to prov e Lemma 3.9 , which states that T ( H ) has at most t wo tight comp onen ts. F or this we will use the follo wing prop osition asserting that H cannot contain 3 v ertices whic h each ha ve a different set of t wo out of three colours. Prop osition 3.8. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 and T ∗ = { T 1 , T 2 , T 3 } b e a set of thr e e distinct tight c omp onents of T ( H ) . Then ther e do not exist thr e e vertic es v 1 , v 2 , v 3 of H such that ϕ ( v i ) = T ∗ \ { T i } for e ach i ∈ [3] . Pr o of. Supp ose for a contradiction that there exist v 1 , v 2 , v 3 ∈ V ( H ) such that ϕ ( v i ) = T ∗ \ { T i } for eac h i ∈ [3]. Cho ose u ∈ N H ( v 1 v 2 ) ∩ N H ( v 2 v 3 ) ∩ N H ( v 1 v 3 ), which exists since | N H ( v 1 v 2 ) ∩ N H ( v 2 v 3 ) ∩ N H ( v 1 v 3 ) | ≥ 3 δ 2 ( H ) − 2 n > 0 . F or all distinct i, j ∈ [3] we hav e ϕ ( uv i v j ) ∈ ϕ ( v i v j ) ⊆ ϕ ( v i ) ∩ ϕ ( v j ) = T ∗ \ { T i , T j } , so T ∗ ⊆ ϕ ( u ), con tradicting Lemma 3.7 . Lemma 3.9. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 . Then T ( H ) has at most two tight c omp onents. Pr o of. Supp ose for a contradiction that T ( H ) has at least 3 distinct tight c omponents. By Prop osi- tion 3.4 and Lemma 3.6(iv) , there exists xy ∈  V ( H ) 2  suc h that | ϕ ( xy ) | = 2. Let ϕ ( xy ) = { T 1 , T 2 } , so ϕ ( x ) = ϕ ( y ) = { T 1 , T 2 } by Lemma 3.7 . Let T 3 b e a tigh t comp onen t in T ( H ) distinct from T 1 or T 2 . By F act 3.3 w e hav e | V ( T 3 ) | > n/ 3, so we may c ho ose z ∈ N H ( xy ) ∩ V ( T 3 ). Since ϕ ( xy z ) ∈ ϕ ( xy ) = { T 1 , T 2 } , w e assume without loss of generalit y that ϕ ( xy z ) = T 1 , so ϕ ( z ) = { T 1 , T 3 } b y Lemma 3.7 . Also by F act 3.3 we hav e deg ∂ T 3 ( z ) > n/ 3, so w e ma y choose w ∈ N ∂ T 3 ( z ) ∩ N H ( y z ), and then w e hav e ϕ ( w y z ) ∈ ϕ ( y z ) = ϕ ( y ) ∩ ϕ ( z ) = { T 1 } . Since T 3 ∈ ϕ ( w z ) and T 2 ∈ ϕ ( xy ), we ma y c ho ose u, v ∈ V ( H ) \ { x, y , z , w } for whic h ϕ ( uw z ) = T 3 and ϕ ( v xy ) = T 2 . This giv es the structure sho wn in Figure 1 . Let P = S e ∈{ v xy,xy z ,wy z ,uw z } ∂ e , so |P | = 9. Since ϕ ( xy z )  = ϕ ( v xy ) and ϕ ( uw z )  = ϕ ( w y z ), by F act 3.1 w e hav e \ P ∈ ∂ xy z ∪ ∂ v xy N H ( P ) = ∅ and \ P ∈ ∂ w y z ∪ ∂ uwz N H ( P ) = ∅ . On the other hand, by Proposition 3.5(i) there exists P ′ ∈ P with T P ∈P \ P ′ N H ( P )  = ∅ . So w e must ha ve P ′ = y z , and therefore ma y choose p ∈ T P ∈P \{ yz } N H ( P ). Since puw z and pv xy are edges of T ( H ) w e ha ve ϕ ( puw ) = ϕ ( uwz ) = T 3 and ϕ ( pv x ) = ϕ ( v xy ) = T 2 , so ϕ ( p ) = { T 2 , T 3 } by Lemma 3.7 . So ϕ ( x ) = { T 1 , T 2 } , ϕ ( p ) = { T 2 , T 3 } and ϕ ( z ) = { T 1 , T 3 } , contradicting Prop osition 3.8 . 9 Figure 1: Edges and comp onen ts in Lemma 3.9 3.3 There is only one tigh t comp onen t In this section we complete the pro of of Lemma 1.4 . By Lemma 3.9 w e kno w that H has only tw o tigh t comp onen ts, whic h w e call red and blue (denoted r and b ), so each edge of H is coloured either red or blue. W e now give sev eral results describing certain colour patterns that cannot o ccur in H . F or edge-coloured 3-graphs G and H , when we write ‘ G is not a subgraph of H ’, we mean that G is not a colour-preserving subgraph of H . Prop osition 3.10. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 , and let r , b b e distinct tight c omp onents of T ( H ) . L et G b e a 3 -gr aph with vertic es x, y 1 , y 2 , y 3 , z and e dge set { xy i y i +1 , z y i y i +1 : i ∈ [3] } , in which 3 e dges have c olour r and 3 e dges have c olour b . Then G is not a sub gr aph of H . Pr o of. Supp ose for a contradiction that H contains a copy of G . Let P b e the set of all pairs which are edges of ∂ G , so |P | = 9. By Prop osition 3.5(i) there exists P ′ ∈ P for whic h T P ∈P \ P ′ N H ( P )  = ∅ . Without loss of generalit y w e ma y assume that x is contained in at least t w o red edges. So either x is in three red edges, in which case w e ma y assume without loss of generality we are in Case 1 b elo w, or x is in precisely tw o red edges, in which case w e may assume without loss of generality w e are in either Case 2 or Case 3 b elo w, according to whether the intersection of the blue edge containing x and the red edge con taining z has size one or tw o. Case 1 : ϕ ( xy 1 y 2 ) = ϕ ( xy 1 y 3 ) = ϕ ( xy 2 y 3 ) = r and y 3 ∈ P ′ . In this case, our choice of P ′ ensures that T P ∈ ∂ xy 1 y 2 ∪ ∂ z y 1 y 2 N H ( P )  = ∅ ; since ϕ ( z y 1 y 2 ) = b this contradicts F act 3.1 . Case 2 : ϕ ( xy 1 y 2 ) = ϕ ( xy 2 y 3 ) = ϕ ( z y 1 y 3 ) = r and ϕ ( z y 1 y 2 ) = ϕ ( z y 2 y 3 ) = ϕ ( xy 1 y 3 ) = b . In this case xy 1 z y 2 is a prop erly coloured C 4 in L ( y 3 ), contradicting Lemma 3.6(i) . Case 3 : ϕ ( xy 1 y 2 ) = ϕ ( xy 2 y 3 ) = ϕ ( z y 1 y 2 ) = r and ϕ ( xy 1 y 3 ) = ϕ ( z y 2 y 3 ) = ϕ ( z y 1 y 3 ) = b . In this case set M = { ( xy 1 y 2 , xy 1 y 3 ) , ( xy 2 y 3 , z y 2 y 3 ) , ( z y 1 y 2 , z y 2 y 3 ) } . Observ e that no pair P ∈ P has P ∈ ∂ e ∪ ∂ e ′ for all three pairs ( e, e ′ ) ∈ M . It follows that some pair ( e, e ′ ) ∈ M has T P ∈ ∂ e ∪ ∂ e ′ N H ( P )  = ∅ . Since eac h pair ( e, e ′ ) ∈ M has | e ∩ e ′ | = 2 and ϕ ( e ) = r  = b = ϕ ( e ′ ), this contradicts F act 3.1 . In the next lemma, we show that H cannot contain a tigh t walk of length 4 with alternating colours (see Figure 2 for an illustration). Lemma 3.11. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 and let r, b b e distinct tight c omp onents of T ( H ) . L et P = v 1 v 2 . . . v 6 b e a tight walk such that ϕ ( v 1 v 2 v 3 ) = ϕ ( v 3 v 4 v 5 ) = r and ϕ ( v 2 v 3 v 4 ) = ϕ ( v 4 v 5 v 6 ) = b . Then P is not a sub gr aph of H . Pr o of. Supp ose for a con tradiction that H contains a copy of P . Observe that the vertices v 2 , v 3 , v 4 and v 5 m ust b e distinct since v 2 v 3 v 4 is a blue edge and v 3 v 4 v 5 is a red edge. Also, since ϕ ( v 1 v 2 v 3 ) = r w e hav e r ∈ ϕ ( v 2 v 3 ) and so by F act 3.3 there are at least n/ 3 vertices u 1 for whic h u 1 v 2 v 3 is a red edge of H . In the same w ay , since ϕ ( v 4 v 5 v 6 ) is a blue edge of H there are at least n/ 3 v ertices u 6 10 Figure 2: A tight path v 1 v 2 . . . v 6 with ϕ ( v 1 v 2 v 3 ) = ϕ ( v 3 v 4 v 5 ) = r and ϕ ( v 2 v 3 v 4 ) = ϕ ( v 4 v 5 v 6 ) = b men tioned in Lemma 3.11 Figure 3: A tight path v 1 v 2 . . . v 7 with ϕ ( v 1 v 2 v 3 ) = ϕ ( v 3 v 4 v 5 ) = ϕ ( v 4 v 5 v 6 ) = r and ϕ ( v 2 v 3 v 4 ) = ϕ ( v 5 v 6 v 7 ) = b mentioned in Lemma 3.12 for whic h v 4 v 5 u 6 is a blue edge of H . By c ho osing such u 1 and u 6 to b e distinct from each other and from v 2 , v 3 , v 4 and v 5 , we obtain a tight path u 1 v 2 v 3 v 4 v 5 u 6 in H with ϕ ( u 1 v 2 v 3 ) = ϕ ( v 3 v 4 v 5 ) = r and ϕ ( v 2 v 3 v 4 ) = ϕ ( v 4 v 5 u 6 ) = b . W e therefore assume without loss of generalit y that P itself is a tigh t path, that is, that v 1 , v 2 , v 3 , v 4 , v 5 and v 6 are all distinct. Let P be the set of pairs which are edges of ∂ P , so |P | = 9. By Prop osition 3.5(i) , there exists P ′ ∈ P such that T P ∈P \ P ′ N H ( P )  = ∅ . If P ′  = v 3 v 4 , then T P ∈ ∂ e ∪ ∂ e ′ N H ( P )  = ∅ for some ( e, e ′ ) ∈ { ( v 1 v 2 v 3 , v 2 v 3 v 4 ) , ( v 3 v 4 v 5 , v 4 v 5 v 6 ) } . By F act 3.1 it follows that e and e ′ are in the same tight comp onen t, contradicting our assumption that ϕ ( e ) = r and ϕ ( e ′ ) = b . W e therefore assume that P ′ = v 3 v 4 . Cho ose v ∈ T P ∈P \ P ′ N H ( P ). Then v v 1 v 2 v 3 and v v 4 v 5 v 6 are copies of K 3 4 in H , so ϕ ( v v 2 v 3 ) = ϕ ( v 1 v 2 v 3 ) = r and ϕ ( v v 4 v 5 ) = ϕ ( v 4 v 5 v 6 ) = b . If ϕ ( v v 2 v 4 ) = r (or ϕ ( v v 3 v 5 ) = b ), then v v 2 v 3 v 5 (or v v 2 v 4 v 5 ) forms a prop erly coloured copy of C 4 in L ( v 4 ) (or in L ( v 3 ) resp ectiv ely), con tradicting Lemma 3.6(i) . So we m ust hav e ϕ ( v v 2 v 4 ) = b and ϕ ( v v 3 v 5 ) = r , and so ϕ ( v v 2 v 3 ) = ϕ ( v 3 v 4 v 5 ) = ϕ ( v v 3 v 5 ) = r and ϕ ( v v 4 v 5 ) = ϕ ( v v 2 v 4 ) = ϕ ( v 2 v 3 v 4 ) = b, giving a con tradiction to Prop osition 3.10 with v 2 , v 5 , v , v 3 and v 4 pla ying the roles of x, z , y 1 , y 2 and y 3 resp ectiv ely . The next lemma forbids an edge-coloured tigh t w alk of length 5 (see Figure 3 for an illustration). Lemma 3.12. L et H b e a 3 -gr aph on n vertic es such that δ 2 ( H ) > 7 n/ 9 and let r, b b e distinct tight c omp onents of T ( H ) . L et P = v 1 v 2 . . . v 7 b e a tight walk such that ϕ ( v 1 v 2 v 3 ) = ϕ ( v 3 v 4 v 5 ) = ϕ ( v 4 v 5 v 6 ) = r and ϕ ( v 2 v 3 v 4 ) = ϕ ( v 5 v 6 v 7 ) = b . Then P is not a sub gr aph of H . Pr o of. Supp ose for a contradiction that H con tains a copy of P . Again the vertices v 2 , v 3 , v 4 and v 5 m ust b e distinct. Since ϕ ( v 5 v 6 v 7 ) = b we ha ve | N b ( v 5 v 6 ) | ≥ n/ 3 by F act 3.3 , so there are at least 11 n/ 3 − 2 n/ 9 = n/ 9 c hoices for y ∈ N ( v 4 v 5 ) ∩ N b ( v 5 v 6 ). By Lemma 3.11 we must hav e ϕ ( v 4 , v 5 ) = { r } , so eac h suc h y has ϕ ( v 4 v 5 y ) = r and ϕ ( v 5 v 6 y ) = b . Having c hosen y , by F act 3.3 there are at least n/ 9 and n/ 3 c hoices for each of z ∈ N b ( v 5 y ) ∩ N H ( v 4 v 5 ) and x ∈ N r ( v 2 v 3 ) resp ectiv ely , so w e may c ho ose such x, y and z to b e distinct from each other and from v 2 , v 3 , v 4 and v 5 , giving a tight path xv 2 v 3 v 4 v 5 y z in H with ϕ ( xv 2 v 3 ) = ϕ ( v 3 v 4 v 5 ) = ϕ ( v 4 v 5 y ) = r and ϕ ( v 2 v 3 v 4 ) = ϕ ( v 5 y z ) = b for which v 4 v 5 z is an edge of H with ϕ ( v 4 v 5 z ) = r . W e therefore assume without loss of generality that P itself is a tigh t path (that is, that v 1 , v 2 , v 3 , v 4 , v 5 , v 6 and v 7 are all distinct) and also that v 4 v 5 v 7 ∈ E ( H ) with ϕ ( v 4 v 5 v 7 ) = r . Let P b e the set of pairs whic h are edges of ∂ P \ { v 1 } , so |P | = 9. By Prop osition 3.5(i) , there exists P ′ ∈ P such that T P ∈P \ P ′ N H ( P )  = ∅ . If P ′  = v 4 v 5 , then T P ∈ ∂ e ∪ ∂ e ′ N H ( P )  = ∅ for some ( e, e ′ ) ∈ { ( v 3 v 4 v 5 , v 2 v 3 v 4 ) , ( v 4 v 5 v 6 , v 5 v 6 v 7 ) } . By F act 3.1 it follows that e and e ′ are in the same tight comp onen t, contradicting our assumption that ϕ ( e ) = r and ϕ ( e ′ ) = b . W e therefore assume that P ′ = v 4 v 5 . Cho ose x ∈ T P ∈P \ P ′ N H ( P ). Then xv 5 v 6 v 7 and xv 2 v 3 v 4 are copies of K 3 4 in H , so ϕ ( xv 5 v 6 ) = ϕ ( v 5 v 6 v 7 ) = b and ϕ ( xv 3 v 4 ) = ϕ ( v 2 v 3 v 4 ) = b . Since v 4 v 6 ∈ P w e know that xv 4 x 6 ∈ E ( H ). So if ϕ ( xv 4 v 6 ) = r , then v 4 v 5 v 6 xv 4 v 3 is a tight w alk in H with ϕ ( v 4 v 5 v 6 ) = ϕ ( xv 4 v 6 ) = r and ϕ ( xv 5 v 6 ) = ϕ ( xv 3 v 4 ) = b . On the other hand, if ϕ ( xv 4 v 6 ) = b , then v 4 v 7 v 5 v 6 v 4 x is a tight w alk in H with ϕ ( v 4 v 5 v 7 ) = ϕ ( v 4 v 5 v 6 ) = r and ϕ ( v 5 v 6 v 7 ) = ϕ ( xv 4 v 6 ) = b . In either case the tight w alk obtained giv es a con tradiction to Lemma 3.11 . By combining the previous tw o lemmas we obtain the follo wing prop osition. Prop osition 3.13. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 such that T ( H ) has pr e cisely two distinct tight c omp onents, r and b . If xy z , w xy , w y z ∈ E ( H ) ar e such that ϕ ( xy z ) = ϕ ( w y z ) = r and ϕ ( w xy ) = b , then ϕ ( z ) = { r } . Pr o of. Supp ose for a contradiction that b ∈ ϕ ( z ), so we may choose v ∈ V ( H ) with b ∈ ϕ ( v z ). By F act 3.3 we may then c ho ose v ′ ∈ N b ( v z ) ∩ N H ( xz ), and in particular we ha ve ϕ ( z v v ′ ) = b . On the other hand, w e m ust ha v e ϕ ( xz v ′ ) = r as otherwise z w y xz v ′ w ould be a tigh t w alk on 6 v ertices whic h con tradicts Lemma 3.11 . So z w y xz v ′ v is a tigh t walk whose colouring contradicts Lemma 3.12 . W e conclude that b / ∈ ϕ ( z ), so ϕ ( z ) = { r } . W e say that a tight comp onent is sp anning in H if every v ertex is in an edge of that tight comp onen t. By Lemma 3.9 w e kno w that T ( H ) has exactly t wo tight comp onen ts, r and b , and F act 3.2 implies that at least one of them is spanning, say r . In the next lemma, we sho w that b is not spanning and study its prop erties. Lemma 3.14. L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) > 7 n/ 9 , let r, b b e distinct tight c omp o- nents of T ( H ) , and supp ose that r is sp anning. Then the fol lowing statements hold. (i) b is not sp anning. (ii) Every xy ∈  V ( H ) 2  with ϕ ( xy ) = { r, b } has N r ( xy ) ⊆ U := { z ∈ V ( H ) : ϕ ( z ) = { r }} . (iii) Ther e is a c opy of K 3 5 in H al l of whose e dges ar e c olour e d b . Pr o of. By Lemma 3.9 , r and b are the only tight comp onents of T ( H ). Fix xy ∈  V ( H ) 2  with ϕ ( xy ) = { r , b } and z ∈ N r ( xy ), so in particular w e ha ve xyz ∈ E ( H ) and ϕ ( xy z ) = r . Prop osition 3.4 ensures that such x and y exist, and also that there exists w ∈ V ( H ) with w xy , w y z ∈ E ( H ) and ϕ ( w xy ) = b . If ϕ ( w y z ) = b then b y Proposition 3.13 (with the roles of r and b swapped, as are the roles of z and w ) we ha ve ϕ ( w ) = { b } , con tradicting our assumption that r is spanning. So we m ust ha ve ϕ ( wy z ) = r , so by Proposition 3.13 w e hav e ϕ ( z ) = { r } . This gives (i) immediately , and also (ii) b ecause z ∈ N r ( xy ) was arbitrary . 12 F or (iii) , observ e that since ϕ ( wxy ) = b , by F act 3.3 w e ma y c ho ose u ∈ N b ( xy ) ∩ N b ( w x ) ∩ N b ( w y ), and then uw xy forms a cop y of K 3 4 in H whose edges are all coloured b . Note that | ∂ w y z ∪ ∂ xy z ∪ ∂ 2 uw xy | = 9 (where ∂ 2 uw xy = { uw , ux, uy , w x, w y , xy } ). So by Prop osition 3.5(i) there exists P ′ ∈ ∂ w y z ∪ ∂ xy z ∪ ∂ 2 uw xy such that \ P ∈ ( ∂ w y z ∪ ∂ xyz ∪ ∂ 2 uwxy ) \ P ′ N H ( P )  = ∅ . If P ′  = y z , then for some ( e, e ′ ) ∈ { ( xy z , uxy ) , ( xy z , wxy ) , ( w y z , uw y ) , ( wy z , w xy ) } , w e ha ve T P ∈ ∂ e ∪ ∂ e ′ N H ( P )  = ∅ , giving a contradiction to F act 3.1 b ecause ϕ ( e ) = r and ϕ ( e ′ ) = b . So we m ust hav e P ′ = y z . Then there exists a vertex u ′ ∈ T P ∈ ∂ 2 uwxy N H ( P ), implying that uu ′ w xy forms a copy of K 3 5 in H . Eac h set of four vertices in this cop y forms a copy of K 3 4 in H which shares at least three vertices with uw xy and so lies in the s ame tight comp onen t b of T ( H ), that is, b . It follo ws that all edges of uu ′ w xy hav e colour b . W e are finally ready to prov e Lemma 1.4 . Pr o of of L emma 1.4 . Suppose for a contradiction that T ( H ) is not tigh tly connected. By Lemma 3.9 , there are then exactly tw o comp onents of T ( H ), say r and b , and b y F act 3.2 and Lemma 3.14(i) w e assume without loss of generality that r is spanning and b is not. Let U := { z ∈ V ( H ) : ϕ ( z ) = { r }} ; b y combining Prop osition 3.4 , F act 3.3 and Lemma 3.14(ii) we then obtain | U | > n/ 3. F urthermore, b y Lemma 3.14(iii) , there is a copy of K 3 5 in H all of whose edges are coloured b ; let y 1 , . . . , y 5 b e the vertices of this copy . F or all distinct i, j ∈ [5] w e hav e | N H ( y i y j ) ∩ U | ≥ δ 2 ( H ) + | U | − n > | U | − 2 n/ 9 > 0 , (3.1) so w e may c ho ose u ij ∈ N H ( y i y j ) ∩ U . By definition of U we then hav e r = ϕ ( u ij y i y j ) ∈ ϕ ( y i y j ), so ϕ ( y i y j ) = { r, b } . By Lemma 3.14(ii) and F act 3.3 it follows that | N H ( y i y j ) ∩ U | = | N r ( y i y j ) | > n/ 3. T ogether with ( 3.1 ) this yields X ij ∈ ( [5] 2 ) | N H ( y i y j ) ∩ U | > 6  | U | − 2 n 9  + 4 n 3 = 6 | U | . By av eraging ov er U it follows that some u ∈ U is in N H ( y i y j ) for at least seven pairs ij ∈  [5] 2  . Applying Man tel’s theorem w e obtain a triangle ij k in the graph on v ertex set [5] whose edges are these pairs. Then uy i y j y k is a copy of K 3 4 in H with ϕ ( y i y j y k ) = b and ϕ ( uy i y j ) = r (because u ∈ U ), giving a contradiction. 4 Hyp ergraph Regularit y In this section, we formulate the notion of hypergraph regularity that w e use, closely follo wing the form ulation from Allen, B¨ ottc her, Co oley and Mycroft [ 1 ]. Recall that a hyp er gr aph H is an ordered pair ( V ( H ) , E ( H )), where E ( H ) ⊆ 2 V ( H ) . W e identify the h yp ergraph H with its edge set E ( H ). A subgraph H ′ of H is a h yp ergraph with V ( H ′ ) ⊆ V ( H ) and E ( H ′ ) ⊆ E ( H ). It is sp anning if V ( H ′ ) = V ( H ). F or U ⊆ V ( H ), we define H [ U ] to b e the subgraph of H with V ( H [ U ]) = U and E ( H [ U ]) = { e ∈ E ( H ) : e ⊆ U } . A h yp ergraph H is called a c omplex if H is down-closed, that is if for an edge e ∈ H and f ⊆ e , then f ∈ H . A k -c omplex is a com plex having only edges of size at most k . W e denote b y H ( i ) 13 the spanning subgraph of H containing only the edges of size i . Given a k -complex H and a set of v ertices S ⊆ V ( H ), the sub complex induced on S is obtained by taking all edges of H induced on S . Let P b e a partition of V ( H ) into vertex classes V 1 , . . . , V s . Then w e say that a set S ⊆ V ( H ) is P -p artite if | S ∩ V i | ≤ 1 for all i ∈ [ s ]. F or P ′ = { V i 1 , . . . , V i r } ⊆ P , w e define the subgraph of H induced b y P ′ , denoted b y H [ P ′ ] or H [ V i 1 , . . . , V i r ], to be the subgraph of H [ S P ′ ] con taining only the edges that are P ′ -partite. The hypergraph H is said to b e P -partite if all of its edges are P -partite. W e say that H is s -p artite if it is P -partite for some partition P of V ( H ) in to s parts. Let H be a P -partite h yp ergraph. If X is a k -set of vertex classes of H , then we write H X for the k -partite subgraph of H ( k ) induced by S X , whose vertex classes are the elemen ts of X . Moreo ver, w e denote b y H X < the k -partite hypergraph with V ( H X < ) = S X and E ( H X < ) = S X ′ ⊊ X H X ′ . In particular, if H is a complex, then H X < is a ( k − 1)-complex b ecause X is a set of size k . Let i ≥ 2 and let P i b e a partition of a vertex set V in to i parts. Let H i and H i − 1 b e a P i - partite i -graph and a P i -partite ( i − 1)-graph on a common vertex set V , resp ectiv ely . W e sa y that a P i -partite i -set in V is supp orte d on H i − 1 if it induces a copy of the complete ( i − 1)-graph K ( i − 1) i on i vertices in H i − 1 . W e denote b y K i ( H i − 1 ) the P i -partite i -graph on V whose edges are all P i - partite i -sets con tained in V whic h are supported on H i − 1 . Now we define the density of H i with r esp e ct to H i − 1 to b e d ( H i | H i − 1 ) = | K i ( H i − 1 ) ∩ H i | | K i ( H i − 1 ) | if | K i ( H i − 1 ) | > 0 and d ( H i | H i − 1 ) = 0 if | K i ( H i − 1 ) | = 0. So d ( H i | H i − 1 ) is the proportion of P i - partite copies of K i − 1 i in H i − 1 whic h are also edges of H i . More generally , if Q = ( Q 1 , Q 2 , . . . , Q r ) is a collection of r (not necessarily disjoint) subgraphs of H i − 1 , we define K i ( Q ) = S r j =1 K i ( Q j ) and d ( H i | Q ) = | K i ( Q ) ∩ H i | | K i ( Q ) | if | K i ( Q ) | > 0 and d ( H i | Q ) = 0 if | K i ( Q ) | = 0. W e say that H i is ( d i , ε, r ) -r e gular with r esp e ct to H i − 1 , if we hav e d ( H i | Q ) = d i ± ε for ev ery r -set Q of subgraphs of H i − 1 with | K i ( Q ) | > ε | K i ( H i − 1 ) | . W e say that H i is ( ε, r ) -r e gular with r esp e ct to H i − 1 if there exists some d i for which H i is ( d i , ε, r )-regular with respect to H i − 1 . Finally , given an i -graph G whose vertex set contains that of H i − 1 , w e sa y that G is ( d i , ε, r ) -r e gular with r esp e ct to H i − 1 if the i -partite subgraph of G induced b y the vertex classes of H i − 1 is ( d i , ε, r )-regular with resp ect to H i − 1 . W e refer to the density of this i -partite subgraph of G with resp ect to H i − 1 as the r elative density of G with r esp e ct to H i − 1 . No w let s ≥ k ≥ 3 and let H b e an s -partite k -complex on v ertex classes V 1 , . . . , V s . F or an y set A ⊆ [ s ], we write V A for S i ∈ A V i . Note that, if e ∈ H ( i ) for some 2 ≤ i ≤ k , then the v ertices of e induce a cop y of K i − 1 i in H ( i − 1) . Therefore, for any set A ∈  [ s ] i  , the density d ( H ( i ) [ V A ] | H ( i − 1) [ V A ]) is the prop ortion of ‘p ossible edges’ of H ( i ) [ V A ], which are indeed edges. W e sa y that H is ( d k , . . . , d 2 , ε k , ε, r ) -r e gular if (a) for an y 2 ≤ i ≤ k − 1 and an y A ∈  [ s ] i  , the induced subgraph H ( i ) [ V A ] is ( d i , ε, 1)-regular with resp ect to H ( i − 1) [ V A ] and (b) for an y A ∈  [ s ] k  , the induced subgraph H ( k ) [ V A ] is ( d k , ε k , r )-regular with respect to H ( k − 1) [ V A ]. F or d = ( d k , . . . , d 2 ), we write ( d , ε k , ε, r )-regular to mean ( d k , . . . , d 2 , ε k , ε, r )-regular. A key prop erty of regular complexes is that the restriction of suc h a complex to a large subset of its vertex set is also a regular complex, with the same relative densities at each level of the com- plex, although with somewhat weak ened regularity prop erties. The next lemma states this prop erty formally . Lemma 4.1 (Regular Restriction Lemma [ 1 , Lemma 28]) . Supp ose inte gers k , m and r e als α, ε, ε k , and d 2 , . . . , d k > 0 ar e such that 1 /m ≪ ε ≪ ε k , d 2 , . . . , d k − 1 and ε k ≪ α, 1 /k . 14 F or any r, s ∈ N and d k > 0 , set d = ( d k , . . . , d 2 ) and let G b e an s -p artite k -c omplex whose vertex classes V 1 , . . . , V s e ach have size m and which is ( d , ε k , ε, r ) -r e gular. Cho ose any V ′ i ⊆ V i with | V ′ i | ≥ α m for e ach i ∈ [ s ] . Then the induc e d sub c omplex G [ V ′ 1 ∪ · · · ∪ V ′ s ] is ( d , √ ε k , √ ε, r ) - r e gular. W e say that a ( k − 1)-complex J is ( t 0 , t 1 , ε ) -e quitable if it has the following prop erties. (a) J is P -partite for some P which partitions V ( J ) into t parts, where t 0 ≤ t ≤ t 1 , of equal size. W e refer to P as the gr ound p artition of J and to the parts of P as the clusters of J . (b) There exists a density ve ctor d = ( d k − 1 , . . . , d 2 ) such that, for each 2 ≤ i ≤ k − 1, we hav e d i ≥ 1 /t 1 and 1 /d i ∈ N and J is ( d , ε, ε, 1)-regular. F or any k -set X of clusters of J , we denote b y ˆ J X the k -partite ( k − 1)-graph ( J X < ) ( k − 1) and call ˆ J X a p olyad . Given a ( t 0 , t 1 , ε )-equitable ( k − 1)-complex J and a k -graph G on V ( J ), we say that G is ( ε k , r ) -r e gular with r esp e ct to a k -set X of clusters of J if there exists some d such that G is ( d, ε k , r )- regular with resp ect to the p oly ad ˆ J X . Moreov er, we write d ∗ G, J ( X ) for the relativ e density of G with resp ect to ˆ J X ; we may drop either subscript if it is clear from con text. W e can no w give the crucial definition of a regular slice. Definition 4.2 (Regular slice) . Given ε, ε k > 0 , r , t 0 , t 1 ∈ N , a k -gr aph G , a ( k − 1) -c omplex J on V ( G ) , is a ( t 0 , t 1 , ε, ε k , r )-regular slice for G if J is ( t 0 , t 1 , ε ) -e quitable and G is ( ε k , r ) -r e gular with r esp e ct to al l but at most ε k  t k  of the k -sets of clusters of J , wher e t is the numb er of clusters of J . If we sp ecify the density vector d and the num b er of clusters t of an equitable complex or a regular slice, then it is not necessary to sp ecify t 0 and t 1 (since the only role of these is to b ound d and t ). In this situation we write that J is ( · , · , ε )-equitable, or is a ( · , · , ε, ε k , r )-regular slice for G . Giv en a regular slice J for a k -graph G , it will b e imp ortant to know the relativ e densities d ∗ ( X ) for k -sets X of clusters of J . T o keep trac k of these we use the following definition. Definition 4.3 (W eigh ted reduced k -graph) . Given a k -gr aph G and a ( t 0 , t 1 , ε ) -e quitable ( k − 1) - c omplex J on V ( G ) , we let R J ( G ) b e the c omplete weighte d k -gr aph whose vertic es ar e the clusters of J and wher e e ach e dge X is given weight d ∗ ( X ) (in p articular, the weight is in [0 , 1] ). When J is cle ar fr om the c ontext we often simply write R ( G ) inste ad of R J ( G ) . Giv en a set S ⊆ V ( G ) of size j for some j ∈ [ k − 1], the r elative de gr e e deg( S ; G ) of S with r esp e ct to G is defined as deg( S ; G ) := |{ e ∈ E ( G ) : S ⊆ e }|  | V ( G ) \ S | k − j  . Similarly , if G is a weigh ted k -graph with w eight function d ∗ , then we define deg( S ; G ) = P e ∈ E ( G ) : S ⊆ e d ∗ ( e )  | V ( G ) \ S | k − j  . Giv en a k -graph G and distinct ‘ro ot’ vertices v 1 , . . . , v ℓ of G and a k -graph H equipp ed with a set of distinct ‘ro ot’ v ertices x 1 , . . . , x ℓ , the numb er of lab el le d r o ote d c opies of H in G , denoted b y n H ( G ; v 1 , . . . , v ℓ ), is defined to b e the n umber of injective maps from V ( H ) to V ( G ) em b edding H in G and taking x j to v j for each j ∈ [ ℓ ]. Then the density of r o ote d c opies of H in G is defined as d H ( G ; v 1 , . . . , v ℓ ) := n H ( G ; v 1 , . . . , v ℓ )  | V ( G ) |− ℓ | V ( H ) |− ℓ  ·  | V ( H ) | − ℓ  ! . This density has a natural probabilistic in terpretation: c ho ose uniformly at random an injective map ψ : V ( H ) → V ( G ) suc h that ψ ( x j ) = v j for each j ∈ [ ℓ ]. Then d H ( G ; v 1 , . . . , v ℓ ) is the probability 15 that ψ embeds H into G . W e no w define H skel to be the ( k − 1)-complex on V ( H ) − ℓ v ertices, obtained from the complex H generated by the down-closure of H by deleting the vertices x 1 , . . . , x ℓ (and all edges containing them) and deleting all the edges of size k . Giv en a ( t 0 , t 1 , ε )-equitable ( k − 1)- complex J on V ( G ), the numb er of r o ote d c opies of H supp orte d by J , written n H ( G ; v 1 , . . . , v ℓ , J ), is defined as the num ber of lab elled ro oted copies of H in G suc h that every v ertex of H skel lies in a distinct cluster of J and the image of H skel is in J (note that we do not require the edges inv olving v 1 , . . . , v ℓ to b e contained in or supp orted b y J and typically they will not b e so). W e also define n ′ H skel ( J ) to b e the n umber of lab elled copies of H skel in J with each vertex of H skel em b edded in a distinct cluster of J . Then the density d H ( G ; v 1 , . . . , v ℓ , J ) of r o ote d c opies of H in G supp orte d by J is defined as d H ( G ; v 1 , . . . , v ℓ , J ) := n H ( G ; v 1 , . . . , v ℓ , J ) n ′ H skel ( J ) . Again there is a natural probabilistic in terpretation: let ψ : V ( H skel ) → V ( G ) b e an injectiv e map c hosen uniformly at random. Extend ψ to a map ψ ′ : V ( H ) → V ( G ) b y taking ψ ′ ( x i ) = v i for eac h i ∈ [ ℓ ]. Then d H ( G ; v 1 , . . . , v ℓ , J ) is the conditional probability that ψ ′ em b eds H in G , giv en that ψ embeds H skel in J with ev ery vertex of H skel em b edded in a different cluster of J . W e now state the statement of the Regular Slice Lemma that w e need, that is a straightforw ard simplification of [ 1 , Lemma 10]. Lemma 4.4 (Regular Slice Lemma [ 1 , Lemma 10]) . L et k ≥ 3 . F or al l p ositive inte gers t 0 , p ositive ε k and al l functions r : N → N and ε : N → (0 , 1] , ther e ar e inte gers t 1 and n 2 such that the fol lowing holds for al l n ≥ n 2 which ar e divisible by t 1 ! . L et G b e a k -gr aph whose vertex set V has size n . Then ther e exists a ( k − 1) -c omplex J on V which is a ( t 0 , t 1 , ε ( t 1 ) , ε k , r ( t 1 )) -r e gular slic e for G such that (i) for e ach i ∈ [ k − 1] , e ach set Y of i clusters of J , we have deg( Y ; R ( G )) = deg ( Y ; G ) ± ε k ; (ii) for e ach 1 ≤ ℓ ≤ 1 /ε k , e ach k -gr aph H e quipp e d with a set of distinct r o ot vertic es x 1 , . . . , x ℓ such that | V ( H ) | ≤ 1 /ε k and any distinct vertic es v 1 , . . . , v ℓ in V , we have   d H ( G ; v 1 , . . . , v ℓ , J ) − d H ( G ; v 1 , . . . , v ℓ )   < ε k . Giv en a regular slice J for a k -graph G , in order to w ork with k -tuples of regular and dense clusters, we define the d -reduced k -graph R J d ( G ) as follows. Definition 4.5 (The d -reduced k -graph) . L et k ≥ 3 . L et G b e a k -gr aph and supp ose J is a ( t 0 , t 1 , ε, ε k , r ) -r e gular slic e for G with t clusters wher e t 0 ≤ t ≤ t 1 . Then, for d > 0 , we de- fine the d -reduced k -graph R J d ( G ) t o b e the k -gr aph whose vertic es ar e the set [ t ] (c orr esp onding to the t clusters of J ) and whose e dges ar e k -sets in  [ t ] k  such that e ∈ E  R J d ( G )  if and only if the c orr esp onding k -set X of clusters of J is such that G is ( ε k , r ) -r e gular with r esp e ct to X and d ∗ ( X ) ≥ d . W e write R d ( G ) for R J d ( G ) when J is clear from con text. The next lemma states that the co degree conditions are also preserved by R d ( G ), so we can work with this structure. Lemma 4.6 ([ 1 , Lemma 12]) . L et G b e a k -gr aph and let J b e a ( · , · , ε, ε k , r ) -r e gular slic e for G with t clusters. F or any set S of at most k − 1 vertic es of R d ( G ) , let S R b e the set of the c orr esp onding at most k − 1 clusters of J . Then we have deg( S ; R d ( G )) ≥ deg( S R ; R ( G )) − d − ζ ( S R ) , wher e ζ ( S R ) is the pr op ortion of k -sets of clusters T with S R ⊆ T which ar e not ( ε k , r ) -r e gular with r esp e ct to G . 16 Giv en a copy of some subgraph H ′ ⊆ H in G ( k ) , how many w ays are there to extend H ′ to a cop y of H in G ( k ) ? The next lemma giv es a low er b ound on this num b er for almost all copies of H ′ in G ( k ) . T o state this precisely , we define the following. Let G b e an s -partite k -complex whose vertex classes V 1 , . . . , V s are eac h of size m and let H b e an s -partite k -complex whose vertex classes X 1 , . . . , X s eac h hav e size at most m . W e say that an em b edding of H in G is p artition-r esp e cting if for an y i ∈ [ s ] the v ertices of X i are em b edded within V i . On the other hand, we sa y G resp ects the partition of H if whenever G contains an i -edge with vertices in V j 1 , . . . , V j i , then H contains an i -edge with vertices in X j 1 , . . . , X j i . W e denote the set of lab elled partition-resp ecting copies of H in G by H G . The Extension Lemma states that if H ′ is an induced sub complex of H and G is regular with G ( k ) reasonably dense, then almost all partition-resp ecting copies of H ′ in G can b e extended to a large n umber of copies of H in G . Lemma 4.7 (Extension Lemma [ 1 , Lemma 29]) . L et k , s, r , b, b ′ , m b e p ositive inte gers, such that b ′ < b and supp ose c, β , d 2 , . . . , d k , ε, ε k b e p ositive c onstants such that 1 /d i ∈ N for any 2 ≤ i ≤ k − 1 and 1 /m ≪ 1 /r, ε ≪ c ≪ ε k , d 2 , . . . , d k − 1 and ε k ≪ β , d k , 1 /s, 1 /b . Supp ose that H is an s -p artite k -c omplex on b vertic es with vertex classes X 1 , . . . , X s and let H ′ b e an induc e d sub c omplex of H on b ′ vertic es. Supp ose that G is an s -p artite k -c omplex with ver- tex classes V 1 , . . . , V s , al l of size m , such that S 0 ≤ i ≤ k − 1 G ( i ) is ( · , · , ε ) -e quitable with density ve ctor ( d k − 1 , . . . , d 2 ) . Supp ose further that for e ach e ∈ H ( k ) with index A ∈  [ s ] k  , the k -gr aph G ( k ) [ V A ] is ( d, ε k , r ) -r e gular with r esp e ct to G ( k − 1) [ V A ] for some d ≥ d k . Then al l but at most β |H ′ G | lab el le d p artition-r esp e cting c opies of H ′ in G extend to at le ast cm b − b ′ lab el le d p artition-r esp e cting c opies of H in G . W e will use the following result due to K ¨ uhn, Mycroft and Osthus [ 18 ]. The maximum vertex de gr e e of a complex G is the maximum degree of a vertex of G . Lemma 4.8 (Em b edding Lemma [ 18 , Lemma 4.5]) . L et ∆ , k , s, r , m 0 b e p ositive inte gers and let c, ε, ε k and d 2 , . . . , d k b e p ositive c onstants such that 1 /d i ∈ N for al l i < k , 1 /m 0 ≪ 1 /r , ε ≪ min { ε k , d 2 , . . . , d k − 1 } ≤ ε k ≪ d k , 1 / ∆ , 1 /s and c ≪ d 2 , . . . , d k . Then the fol lowing holds for al l inte gers m ≥ m 0 . Supp ose that G is an s -p artite k -c omplex of maximum vertex de gr e e at most ∆ with vertex classes X 1 , . . . , X s such that | X i | ≤ cm for e ach i ∈ [ s ] . Supp ose also that H is a ( d , ε k , ε, r ) -r e gular s -p artite k -c omplex with vertex classes V 1 , . . . , V s al l of size m , r esp e cting the p artition of G . Then H c ontains a lab el le d p artition-r esp e cting c opy of G . W e now sho w that if H is a 3-graph with δ 2 ( H ) ≥ ( γ + α ) n , then the reduced graph ‘almost’ inherits the co degree condition. W e b orrow the follo wing terminology from Han, Lo and Sanh ueza- Matamala [ 11 ] to describ e this. F or 0 ≤ µ, θ ≤ 1 and a k -graph H on n v ertices, H is said to b e ( µ, θ ) -dense if there exists S ⊆  V ( H ) k − 1  of size at most θ  n k − 1  suc h that for all S ∈  V ( H ) k − 1  \ S w e ha ve deg H ( S ) ≥ µ ( n − k + 1). Prop osition 4.9. L et 1 /t ≪ d 3 , ε 3 ≪ α, β and 0 ≤ γ ≤ 1 . L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ ( γ + α ) n and J b e a ( · , · , ε, ε 3 , r ) -r e gular slic e for H with t clusters such that for any set Y of k − 1 clusters of J , we have deg ( Y ; R ( H )) = deg ( Y ; H ) ± ε 3 . L et R = R J d 3 ( H ) . Then at le ast (1 − β )  t 2  p airs of vertic es in R have c o de gr e e at le ast ( γ + α/ 2) t , i.e. R is ( γ + α / 2 , β ) -dense. 17 Pr o of. Let V 1 , V 2 ∈ V ( R ( H )), so V 1 and V 2 are clusters of J . W e hav e deg( V 1 V 2 ; R ( H )) = deg( V 1 V 2 ; H ) ± ε 3 ≥ γ + 3 α/ 4 . Since J is a regular slice, H is ( ε 3 , r )-regular with resp ect to all but at most ε 3  t 3  triples of clusters of J . So at most 1 2 √ ε 3  t 2  pairs of clusters of J lie in at least 2 √ ε 3 t of irregular triples. Call a pair of clusters in R ( H ) ‘go o d’ if they lie in less than 2 √ ε 3 t irregular triples. The n umber of go o d pairs of clusters of J is at least (1 − 1 2 √ ε 3 )  t 2  ≥ (1 − β )  t 2  . F or i ∈ [ t ], let i b e the vertex in R for which V i is the corresp onding cluster of J . Lemma 4.6 implies that for any go od pair of clusters V i , V j ∈ R ( H ), w e hav e deg( ij ; R ) ≥  deg( V i V j ; R ( H )) − d 3 − 2 √ ε 3 t t − 2  ( t − 2) ≥  γ + α 2  t. This shows that all but at most β  t 2  pairs of vertices ha ve co degree at least ( γ + α/ 2) t in R . Sa y that a 3-graph H is str ongly ( µ, θ ) -dense if it is ( µ, θ )-dense and for all edges e ∈ E ( H ) and all pairs X ⊆ e w e hav e deg H ( X ) ≥ µ ( n − 2). W e use the following result by Han, Lo and Sanh ueza-Matamala [ 11 ]. Their statement of the lemma actually applied to all k ≥ 3, and did not include the final “moreov er” statement, but this can b e read off from their pro of. Lemma 4.10 ([ 11 , Lemma 8.4 for k = 3]) . L et n ≥ 6 and 0 < µ, θ < 1 . L et H b e a 3 -gr aph on n vertic es that is ( µ, θ ) -dense. Then ther e exists a sub gr aph H ′ on V ( H ) that is str ongly ( µ − 8 θ 1 / 4 , θ + θ 1 / 4 ) -dense. Mor e over, | E ( H − H ′ ) | ≤ 8 θ 1 / 2  n 3  . 5 Connecting Lemma The main aim of this section is to prov e the Connecting Lemma (Lemma 1.5 ). A rough sketc h of the pro of of the Connecting Lemma is as follows. In the pro of of Lemma 1.5 , w e use the strong Hy- p ergraph Regularity , particularly , the Regular Slice Lemma (Lemma 4.4 ) prov ed b y Allen, B¨ ottcher, Co oley and Mycroft [ 1 ]. As a consequence of the Regular Slice Lemma, an appropriate reduced 3- graph of H inherits the minimum co degree condition approximately . In order to conv ert this to a minimum co degree condition, we use a result (Lemma 5.1 ) b y F erb er and Kwan [ 8 ]. Once w e ha ve a minimum co degree in the reduced 3-graph, w e use Lemma 1.4 to get a squared-tigh t-walk b et ween any t wo triples of vertices of the reduced 3-graph. Note that a tight w alk lets us fix the order of the initial triple but not the final one. In order to hav e a squared-tigh t-walk that resp ects the order of b oth the initial and final triples of v ertices, w e preserv e a K 3 5 in the reduced graph. W e then use the Extension Lemma (Lemma 4.7 ) due to Allen, B¨ ottcher, Co oley and Mycroft to get a squared-tigh t-path b et ween almost all pairs of ordered vertex-triples supp orted on the regular slice (see Section 4 for the definition of a regular slice) on a constant num b er of vertices. Each pair of v ertex-triples ( e 1 , e 2 ) in H lo cally extends to one such pair of ordered vertex-triples supp orted on the regular slice, av oiding the small forbidden set X . The next result by F erb er and Kwan [ 8 ] pro vides a wa y of translating an ‘almost minim um co degree’ condition to an exact minimum co degree condition on a suitable subgraph. Lemma 5.1 ([ 8 , Lemma 3.4]) . L et G b e a k -gr aph on n vertic es in which at most δ  n ℓ  sets X ∈  V ( G ) ℓ  have deg G ( X ) < ( µ + η )  n − ℓ k − ℓ  . Also let Q ≥ 2 ℓ , and cho ose a set S ⊆ V ( G ) of size | S | = Q uniformly at r andom. Then with pr ob ability at le ast 1 −  Q ℓ  ( δ + e − η 2 Q/ 4 k 2 ) the induc e d sub gr aph G [ S ] has minimum ℓ -de gr e e δ ℓ ( G [ S ]) ≥ ( µ + η / 2)  Q − ℓ k − ℓ  . W e first find a short squared-tight-w alk b et ween an y t wo ordered triples of v ertices that are edges of an appropriate reduced graph, resp ecting the order of b oth triples. 18 Lemma 5.2. L et 1 /t ≪ η , β ≪ α . L et R b e a 3 -gr aph on t vertic es that is str ongly (7 / 9 + α, β ) -dense, and let x and y b e or der e d triples which ar e e dges of R . Then ther e exists a squar e d-tight-walk on at most 1 /η vertic es fr om x to y . Pr o of. Fix T suc h that 1 /t ≪ η , β ≪ 1 /T ≪ α . Let W ⊆ V ( R ) b e a set of T v ertices chosen uniformly at random. By Lemma 5.1 with R, 3 , t, T , β , 7 / 9 , α, 2 pla ying the roles of G, k , n, Q, δ, µ, η , ℓ , resp ectiv ely , P  δ 2 ( R [ W ]) ≥  7 9 + α 2  T  ≥ 1 −  T 2   β + e − α 2 T / 36  ≥ 99 100 . Note that for all x, y ∈ V ( R ) with deg R ( xy ) > 0, E deg R ( xy , W ) = (7 / 9 + α ) | W | and deg R ( xy , W ) ∼ Hyp( t, T , deg R ( xy )). By Ho effding’s inequalit y (Lemma 2.5 ), we hav e P (deg R ( xy , W ) ≤ (7 / 9 + α / 2) T ) ≤ 2 e − 7 α 2 T / 108 . T aking a union b ound ov er all elements of { xy ∈  V ( H ) 2  : deg R ( xy ) > 0 } , we deduce that with high probabilit y , we ha v e δ 2 ( R [ W ]) ≥ (7 / 9 + α / 2) T and for all xy ∈  V ( R ) 2  suc h that deg R ( xy ) > 0, deg R ( xy , W ) ≥ (7 / 9 + α/ 2) T . Fix such a set W of v ertices. Note that by Theorem 2.2 , W contains a K 3 5 . Let V  K 3 5  = { v i : i ∈ [5] } . W e no w connect x and y via W . Let x = x 1 x 2 x 3 , y = y 1 y 2 y 3 and x 4 , x 5 , x 6 ∈ W \ { x i , y i : i ∈ [3] } b e distinct vertices such that x 1 x 2 x 3 x 4 x 5 x 6 is a squared-tight-path. There are at least ( T / 3) 3 w ays to pick x 4 , x 5 , x 5 ∈ W \ { x i , y i : i ∈ [3] } . Similarly , fix y 4 , y 5 , y 6 ∈ W \ { x i , x i +3 , y i : i ∈ [3] } suc h that y 3 y 2 y 1 y 4 y 5 y 6 is a squared-tight-path. By Lemma 1.4 , T ( R [ W ]) is tigh tly connected. Th us, there exists a squared-tight-w alk W 1 from x 4 x 5 x 6 to v σ (1) v σ (2) v σ (3) for some σ ∈ S 3 . Similarly , there exists a squared-tigh t-walk W 2 from y 4 y 5 y 6 to v σ ′ (1) v σ ′ (2) v σ ′ (3) for some σ ′ ∈ S 3 . Let W ′ b e a squared- tigh t-walk on { v i : i ∈ [5] } from v σ (1) v σ (2) v σ (3) to v σ ′ (3) v σ ′ (2) v σ ′ (1) . Let ← − W 2 b e the squared-tight-w alk obtained from W 2 b y reversing its vertex sequence. Let W = W 1 W ′ ← − W 2 b e the squared-tight-w alk from x 4 x 5 x 6 to y 6 y 5 y 4 . Note that by deleting all vertices in W b et w een rep eated o ccurrences of an ordered triple of v ertices, we ma y assume that W con tains at most T 3 v ertices. Th us, x 1 x 2 x 3 W y 1 y 2 y 3 is a squared- tigh t-walk from x to y on at most T 3 + 6 ≤ 1 /η vertices. W e now find a short squared-tigh t-path b et ween an y tw o ordered triples of v ertices that are edges of H , resp ecting the order of b oth triples. Lemma 5.3. L et 1 /n ≪ η ≪ α . L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n , and let x and y b e or der e d triples which ar e e dges of H . Then ther e is a squar e d-tight-p ath in H on at most 1 /η vertic es fr om x to y . Pr o of. Cho ose constants: 1 /n ≪ 1 /t 1 ≪ 1 /t 0 ≪ 1 /r , ε ≪ c ≪ ε 3 ≪ β ′ ≪ d 2 , d 3 , η ≪ θ , β ≪ α. Let η ′ = ( ⌊ 1 /η ⌋ − 6) − 1 , x = x 1 x 2 x 3 and y = y 1 y 2 y 3 . Let H ′ b e an induced subgraph of H on n ′ v ertices by deleting at most t 1 ! arbitrary v ertices from V ( H ) \ { x i , y i : i ∈ [3] } , suc h that n ′ ≡ 0 (mo d t 1 !). Note that δ 2 ( H ′ ) ≥ (7 / 9 + α ) n − t 1 ! ≥ (7 / 9 + 5 α/ 6) n ′ . By Lemma 4.4 with H ′ , n ′ , 3 pla ying the roles of G, n, k , resp ectively , let J b e the ( · , · , ε, ε 3 , r )-regular slice for H ′ on t 0 ≤ t ≤ t 1 clusters V 1 , . . . , V t , densit y parameter d 2 and let R = R d 3 ( H ′ ). By Prop osition 4.9 with H ′ , n ′ , 5 α/ 6 , θ , 7 / 9 pla ying the roles of H , n, α , β , γ , resp ectively , R is (7 / 9 + 5 α/ 12 , θ )-dense. By Lemma 4.10 , with R and 7 / 9 + 5 α / 12 pla ying the roles of H and µ resp ectively , there exists a subgraph R ′ of R which is strongly (7 / 9 + α/ 3 , 2 θ 1 / 4 )-dense. Moreov er, | E ( R − R ′ ) | ≤ 8 √ θ  t 3  . 19 W e say that the ordered triple x 4 x 5 x 6 of v ertices is an extension of the ordered triple x 1 x 2 x 3 if x 1 x 2 x 3 x 4 x 5 x 6 is a squared-tight-path in H ′ . for ij k ∈ E ( R ), let J ij k = { v i v j v k ∈ V i × V j × V k : { v i , v j , v k } ∈ K 3 ( J ) } . Note that ij k and v i v j v k are considered as ordered triples. Claim 5.4. L et I ⊆ [ t ] b e such that | I | ≤ 3 . Then ther e exists an or der e d triple ij k ∈  V ( R ′ ) \ I 3  and J ′ ij k ⊆ J ij k such that ij k ∈ E ( R ′ ) , |J ′ ij k | ≥ |J ij k | / 27 and e ach memb er of J ′ ij k is an extension of x 1 x 2 x 3 . Pr o of of claim. Let ˜ H b e a squared-tigh t-path on 6 v ertices from x 1 x 2 x 3 . Observ e that by the co degree of H ′ and F act 3.3 , the n umber of ordered triples x 4 x 5 x 6 suc h that x 1 . . . x 6 forms an extension of x 1 x 2 x 3 in H ′ is at least  1 3 + 5 α 2  n ′  1 3 + 5 α 2  n ′ − 1   1 3 + 5 α 2  n ′ − 2  ≥  1 27 + α  n ′ 3 . By Lemma 4.4(ii) with ˜ H , 3 , ε 3 pla ying the role of H , ℓ, ε k , resp ectively , we hav e d ˜ H ( H ′ ; x 1 , x 2 , x 3 , J ) ≥ d ˜ H ( H ′ , x 1 , x 2 , x 3 ) − ε 3 ≥ 1 / 27 + α − ε 3 ≥ 1 / 27 + α/ 2 . By deleting the prop ortion of triples of clusters which do not corresp ond to edges of R ′ and those in volving I , w e are left with at least 1 / 27 + α / 2 − d 3 − ε 3 − 8 √ θ − 3 /t ≥ 1 / 27 + α/ 4 prop ortion of the triples of v ertices. Eac h of these triples of v ertices is supported on triples of clusters whic h corresp ond to edges in R ′ \ I . By av eraging ov er all edges in R ′ \ I , there exists ij k ∈ E ( R ′ ) \ I suc h that d ˜ H ( H ′ ; x 1 , x 2 , x 3 , J ij k ) ≥ 1 / 27 + α/ 4 . Let J ′ ij k = { v i v j v k ∈ J ij k : v i v j v k is an extension of x 1 x 2 x 3 } . Therefore, w e hav e |J ′ ij k | ≥ |J ij k | / 27.  Let ij k and J ′ ij k b e giv en b y Claim 5.4 with I = ∅ . Applying Claim 5.4 again with I = { i, j, k } and x 1 x 2 x 3 = y 3 y 2 y 1 (as an ordered triple), there exists an ordered triple i ′ j ′ k ′ ∈ E ( R ′ ) with ij k ∩ i ′ j ′ k ′ = ∅ and J ′ i ′ j ′ k ′ ⊆ J i ′ j ′ k ′ suc h that for all ordered triples v i ′ v j ′ v k ′ ∈ J ′ i ′ j ′ k ′ , we hav e v i ′ v j ′ v k ′ is an extension of y 3 y 2 y 1 and |J ′ i ′ j ′ k ′ | ≥ |J i ′ j ′ k ′ | / 27. By Lemma 5.2 with R ′ , α/ 3 , 2 θ 1 / 4 , η ′ , ij k , k ′ j ′ i ′ pla ying the roles of R , α, β , η , x , y , resp ectiv ely , there exists a squared-tight-w alk W in R ′ of length at most 1 /η ′ from ij k to k ′ j ′ i ′ . Let W = w 1 w 2 . . . w b where b ≤ 1 /η ′ , w 1 w 2 w 3 = ij k , w b − 2 w b − 1 w b = k ′ j ′ i ′ . Without loss of generality , we may assume the distinct vertices app earing in W are [ s ] and when coun ted with m ultiplicity , there are b v ertices. Note that W is s -partite and 6 ≤ s ≤ b ≤ 1 /η ′ . Let H (3) b e the squared-tight-path formed b y replacing each rep eated vertex of W with distinct copies of it, whenever it is rep eated in W . Note that H (3) is s -partite with vertex classes V 1 , . . . , V s where { V i } i ∈ [ b ] is the set of copies of the v ertex classes of W . Let H b e the do wn-closure of H (3) , so H is an s -partite 3-complex. Let H ′ b e the sub complex induced on the first and final three v ertices of the squared-tight-path, i.e. on the vertex classes corresp onding to ij k and k ′ j ′ i ′ . Let G b e the s -partite 3-complex obtained from J h S p ∈ [ s ] V p i b y adding the edges of H ′ supp orted on J ′ ij k and J ′ i ′ j ′ k ′ for the triples of vertices w 1 w 2 w 3 = ij k and w b − 2 w b − 1 w b = k ′ j ′ i ′ , the edges supp orted on J h S p ∈ [ b − 2] V w p V w p +1 V w p +2 i for all v ertex triples { w p w p +1 w p +2 : p ∈ [ b − 3] \ { 1 }} and all the edges of H ′ supp orted on J for an y other triple. Note that for any edge e ∈ H (3) with 20 index A ∈  [ s ] 3  , the 3-graph G (3) [ V A ] is either ( d, ε 3 , r )-regular with d ≥ d 3 (on the triples of clusters that are edges of W ) or (1 , ε 3 , r )-regular (on any other triple) with resp ect to G (2) [ V A ]. All the conditions to apply Lemma 4.7 are thus satisfied. Consider a pair of ordered triples of v ertices, v 1 v 2 v 3 from V w 1 V w 2 V w 3 and v b − 2 v b − 1 v b from V w b − 2 V w b − 1 V w b . Note that by our assumption on ij k and i ′ j ′ k ′ to b e v ertex-disjoin t, so are v 1 v 2 v 3 and v b − 2 v b − 1 v b . By Lemma 4.7 with 3 , 6 , β ′ pla ying the roles of k, b ′ , β , resp ectiv ely , for all but at most β ′ prop ortion of all ordered pairs of triples of v ertices ( v 1 v 2 v 3 , v b − 2 v b − 1 v b ) from the clusters of H ′ , there are at least cm b − 6 ≥ cm 3 extensions to lab elled partition-resp ecting copies of H in G . Each such cop y of H corresp onds to a squared- tigh t-path in H ′ on b v ertices with every v ertex in the required cluster. Since β ′ ≪ d 2 , |J ′ ij k | ≥ |J ij k | / 27 and |J ′ i ′ j ′ k ′ | ≥ |J ′ i ′ j ′ k ′ | / 27 by Claim 5.4 , at least one suc h copy of H (3) is a squared-tight- path going from an extension of x 1 x 2 x 3 to an extension of y 3 y 2 y 1 . Let P b e one such squared- tigh t-path. Then x 1 x 2 x 3 P y 1 y 2 y 3 is the required squared-tight-path in H from x to y on at most b + 6 ≤ 1 /η ′ + 6 ≤ 1 /η vertices. W e are finally ready to prov e Lemma 1.5 . Pr o of of L emma 1.5 . Let E := { x i , y i : i ∈ [ s ] } , and for eac h i ∈ [ s ] let X i := ( X ∪ V ( E )) \ ( V ( x i ) ∪ V ( y i )). Let ℓ ∈ [ s ] b e maximal with the prop erty that there exists a collection P = { P 1 , . . . , P ℓ } of v ertex-disjoint squared-tight-paths such that for eac h i ∈ [ ℓ ] the path P i has initial triple x i , final triple y i , order | V ( P i ) | ≤ ψ − 1 / 2 and V ( P i ) ∩ X i = ∅ . Supp ose for a con tradiction that ℓ < s . Let X ′ := X ℓ +1 ∪ S i ∈ [ ℓ ] V ( P i ), so | X ′ | ≤ | X | +6 s + ψ − 1 / 2 ℓ ≤ ψ n + 6 ψ n + √ ψ n ≤ 8 √ ψ n . Also let H ′ := H \ X ′ and n ′ := | V ( H ′ ) | , so δ 2 ( H ′ ) ≥ (7 / 9 + α ) n − 8 √ ψ n ≥ (7 / 9 + α/ 2) n ′ . Lemma 5.3 , with H ′ , n ′ , α/ 2 , x ℓ +1 , y ℓ +1 and √ ψ pla ying the roles of H , n, α, x , y and η resp ectively , implies that there exists a squared-tigh t-path P ℓ +1 in H ′ from x ℓ +1 to y ℓ +1 on at most ψ − 1 / 2 v ertices. The collection P ∪ { P ℓ +1 } then contradicts the maximality of ℓ . W e conclude that ℓ = s , and the collection P is as desired, since | V ( P ) | ≤ ψ − 1 / 2 s ≤ √ ψ n . 6 Absorption The aim of this section is to prov e Lemma 1.6 . Our approach loosely follo ws the metho ds used by Ara ´ ujo, Piga and Schac h t [ 2 ] and H` an, P erson and Schac h t [ 10 ], who in turn follow ed the work of R¨ odl, Ruci ´ nski and Szemer´ edi (see, e.g. [ 24 , 25 ]). W e first formally define an absorb er for our problem. Let H b e a 3-graph on v ertex set V and let ( v 1 , v 2 , v 3 , v 4 ) ∈ V 4 . A lab elled set of 36 vertices A = { x i , y i , z i , u i,j : i ∈ [4] , j ∈ [6] } is an absorb er for v := ( v 1 , v 2 , v 3 , v 4 ) if the ordered sequences (i) T 1 ( A ) := x 1 . . . x 4 y 1 . . . y 4 z 1 . . . z 4 , (ii) T 2 ( A ) := x 1 . . . x 4 z 1 . . . z 4 , (iii) U i ( A, v ) := u i, 1 u i, 2 u i, 3 v i u i, 4 u i, 5 u i, 6 for each i ∈ [4] and (iv) U i ( A ) := u i, 1 u i, 2 u i, 3 y i u i, 4 u i, 5 u i, 6 for each i ∈ [4] are eac h squared-tight-paths in H . W e say that A is an absorb er if it is an absorb er for some ( v 1 , v 2 , v 3 , v 4 ) ∈ V 4 . W e also write f A, v for the function whic h maps T 2 ( A ) to T 1 ( A ) and, for eac h i ∈ [4], maps U i ( A ) to U i ( A, v ). The purp ose of this definition is the following. Supp ose A is an absorb er for v := ( v 1 , v 2 , v 3 , v 4 ), where v 1 , v 2 , v 3 and v 4 are distinct vertices of H . Then P A := { T 2 ( A ) , U 1 ( A ) , U 2 ( A ) , U 3 ( A ) , U 4 ( A ) } is a collection of fiv e vertex-disjoin t squared-tight-paths with vertex set V ( P A ) = A . Moreov er, Q A, v := { f A, v ( P ) : P ∈ P A } = { T 1 ( A ) , U 1 ( A, v ) , U 2 ( A, v ) , U 3 ( A, v ) , U 4 ( A, v ) } 21 is a collection of fiv e v ertex-disjoin t squared-tigh t-paths with vertex set V ( Q A, v ) = A ∪ { v 1 , v 2 , v 3 , v 4 } , and each path in Q A, v has the same ends as the corresponding path in P A . So the effect of replacing eac h path P ∈ P A b y f A, v ( P ) is to ‘absorb’ the vertices v 1 , v 2 , v 3 and v 4 in to the squared-tight-paths of the absorb er. W e first show that, for each ( v 1 , v 2 , v 3 , v 4 ) ∈ V 4 , a constan t prop ortion of all lab elled sets of 36 v ertices of H are absorbers for ( v 1 , v 2 , v 3 , v 4 ). Recall that, for a 3-graph F , γ ( F ) is the codegree T ur´ an density of F . Lemma 6.1. L et 1 /n ≪ c ≪ α , let H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n , and let v 1 , v 2 , v 3 and v 4 b e vertic es of H . Then ther e ar e at le ast cn 36 absorb ers for ( v 1 , v 2 , v 3 , v 4 ) in H . Pr o of. Let β = (2 c ) 1 / 5 . By a greedy argumen t analogous to F act 3.3 , w e hav e γ ( K 3 4 ) ≤ 2 / 3. Since δ 2 ( H \ { v 1 , v 2 , v 3 , v 4 } ) > 7( n − 4) / 9, by Theorem 2.1 , there are at least β n 12 copies of K 3 4 (3) in H a voiding v 1 , v 2 , v 3 , v 4 . Fix a cop y of a K 3 4 (3) whose v ertex classes are V i = { x i , y i , z i } for eac h i ∈ [4]. Note that x 1 . . . x 4 y 1 . . . y 4 z 1 . . . z 4 and x 1 . . . x 4 z 1 . . . z 4 are b oth squared-tigh t-paths in H . Fix i ∈ [4] and consider L i = L ( v i ) ∩ L ( y i ) \ { v 1 , v 2 , v 3 , v 4 } . Note that L i is a 2-graph on n − 5 v ertices with δ ( L i ) ≥ 2(7 / 9 + α ) n − n − 3 ≥ (5 / 9 + α ) n. (6.1) This implies that for an y edge ab ∈ E ( L i ), we hav e | N L i ( a ) ∩ N L i ( b ) | ≥ 2(5 / 9 + α ) n − n ≥ (1 / 9 + 2 α ) n. The num b er of triangles in L i is therefore at least 1 3 | E ( L i ) |  1 9 + 2 α  n ≥ 5 81  n 3  . (6.2) W e say an unordered triple of v ertices w 1 , w 2 , w 3 in L i is a nic e triple if w 1 w 2 w 3 is a triangle in L i as w ell as w 1 w 2 w 3 ∈ E ( H ). W e now show that there is a set of nice triples corresp onding to ev ery triangle in L i . Let a 1 a 2 a 3 b e a triangle in L i . Observe that n ≥       [ j ∈ [3] N L i ( a j )       ≥ X j ∈ [3] | N L i ( a j ) | − X j k ∈ ( [3] 2 ) | N L i ( a j ) ∩ N L i ( a k ) | , implying that X j k ∈ ( [3] 2 ) | N L i ( a j ) ∩ N L i ( a k ) | ( 6.1 ) ≥ 3(5 / 9 + α ) n − n ≥ (2 / 3 + 3 α ) n. By av eraging, w e may assume without loss of generality , that | N L i ( a 1 ) ∩ N L i ( a 2 ) | ≥ (2 / 9 + α ) n . Using the co degree of H we deduce that | N H ( a 1 a 2 ) ∩ N L i ( a 1 ) ∩ N L i ( a 2 ) | ≥ (2 / 9 + α ) n − (2 / 9 − α ) n = 2 αn. Therefore, there is a set B ⊆ V ( L i ) of at least 2 αn vertices suc h that for all b ∈ B , we ha ve a 1 a 2 b is a triangle in L i and a 1 a 2 b ∈ E ( H ). F or each triangle in L i , we thus obtain at least 2 α n nice triples. Observ e that each nice triple can b e obtained from at most 3 n triangles in L i . So b y ( 6.2 ) there are at least 1 3 n  5 81  n 3  2 αn ≥ αn 3 200 nice triples of vertices in L i . 22 Let H ′ b e an auxiliary 3-graph on vertex set V ( H ) whose edges are nice triples of L i . Since there is a p ositive density of edges in H ′ and the T ur´ an densit y of an edge is 0, by Theorem 2.1 , there are at least β n 6 2-blo wups of edges of H ′ . Observ e that a 2-blo wup of an edge in H ′ is a set of 6 v ertices u i, 1 , u i, 2 , . . . , u i, 6 suc h that u i, 1 . . . u i, 6 is a squared path in L ( v i ) ∩ L ( y i ) and furthermore u i,j u i,j +1 u i,j +2 ∈ E ( H ) for all j ∈ [4]. By repeating this argument for eac h i ∈ [4], we obtain at least β 5 n 36 c hoices for an absorb er A for ( v 1 , v 2 , v 3 , v 4 ). The only wa y this could fail to b e an absorb er is if the 36 vertices obtained are not all distinct. Removing at most 36 2 n 35 man y suc h sets where the vertices are not distinct, we obtain at least β 5 n 36 / 2 = cn 36 absorb ers for ( v 1 , v 2 , v 3 , v 4 ). This prov es the result. W e say t wo absorbers A 1 and A 2 are disjoint if the underlying (unlab elled) sets are disjoin t, and otherwise that A 1 and A 2 interse ct . In the next lemma we use a random selection to obtain a linear-size family F of pairwise-disjoint absorb ers with the prop ert y that ev ery 4-tuple of v ertices of H has man y absorb ers in F . Lemma 6.2. L et 1 /n ≪ β ≪ α and let H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n . Then ther e exists a family F of at most β n p airwise-disjoint absorb ers such that for e ach ( v 1 , v 2 , v 3 , v 4 ) ∈ V ( H ) 4 ther e ar e at le ast β 2 n absorb ers for ( v 1 , v 2 , v 3 , v 4 ) in F . Pr o of. Let 1 /n ≪ β ≪ c ≪ α . F or each 4-tuple of v ertices v = ( v 1 , . . . .v 4 ) ∈ V ( H ) 4 , let A ( v ) denote the set of absorb ers for v , so |A ( v ) | ≥ cn 36 b y Lemma 6.1 . Cho ose a family F ′ of lab elled sets of size 36 by selecting each of the  n 36  36! lab elled sets with probability p = β / (2 n 35 ), indep endently of all other choices. Then |F ′ | is a binomial random v ariable with exp ectation E |F ′ | = p  n 36  36! ≤ β n 2 , so by Chernoff ’s inequality (Lemma 2.4 ) we hav e |F ′ | ≤ β n with probability at least 3 / 4. Similarly , for each v ∈ V ( H ) 4 the quantit y |A ( v ) ∩ F ′ | is a binomial random v ariable with E |A ( v ) ∩ F ′ | ≥ cn 36 p = cβ n 2 . So b y applying Chernoff ’s inequality (Lemma 2.4 ) and taking a union b ound ov er all v ∈ V ( H ) 4 , with probability at least 3/4 it holds that for every v ∈ V ( H ) 4 w e hav e |A ( v ) ∩ F ′ | ≥ cβ n/ 4. Finally , the exp ected num b er of pairs of absorb ers that intersect is at most n 36 36 2 n 35 p 2 = 36 2 β 2 n/ 4 . So b y Mark ov’s inequalit y (Lemma 2.3 ), with probability at least 3 / 4 it holds that F ′ con tains at most 36 2 β 2 n pairs of intersecting absorb ers. Fix a choice of F ′ for which each of these three even ts o ccurs, and let F b e the subfamily of F ′ formed b y deleting from F ′ b oth mem b ers of eac h in tersecting pair of absorbers and also each F ∈ F ′ whic h is not an absorb er. Then F is a family of at most β n pairwise-disjoin t absorbers with the prop ert y that |A ( v ) ∩ F | ≥ cβ n/ 4 − 2(36 β ) 2 n ≥ β 2 n for each v ∈ V ( H ) 4 . By choosing a family of absorb ers as in Lemma 6.2 we obtain a collection of pairwise v ertex-disjoint squared-tigh t-paths which can absorb any small set of v ertices whose size is a multiple of four (this can b e seen b y follo wing the pro of of the next lemma, ignoring the path ˆ P and taking L ′ = L ). T o b e able to absorb sets whose size is not a multiple of four, w e also use a copy of K 3 5 (4) in H , which ma y ‘donate’ up to three vertices to the set we w ant to absorb. T o b e more sp ecific we introduce the follo wing notation: for a vertex sequence P = v 1 v 2 . . . v ℓ and a subset S ⊆ V ( P ), we write P \ S to denote the sequence of v ertices of V ( P ) \ S in the same order as in P . Observe that if a cop y of K 3 5 (4) in H has vertex classes { a i , b i , c i , d i } for eac h i ∈ [4], then P := a 1 . . . a 5 b 1 . . . b 5 c 1 . . . c 5 d 1 . . . d 5 is a squared-tigh t-path in H with the prop ert y that, for any subset S ⊆ { a 5 , b 5 , c 5 } , P \ S is a squared- tigh t-path in H with the same ends as P . 23 Lemma 6.3. L et 1 /n ≪ β ≪ α . L et H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (7 / 9 + α ) n . Then ther e exists a c ol le ction P of p airwise vertex-disjoint squar e d-tight-p aths in H with |P | ≤ 5 β n + 1 , with | V ( P ) | ≤ 40 β n , and with the fol lowing absorbing pr op erty. F or al l sets L ⊆ V ( H ) \ V ( P ) with | L | ≤ β 2 n , ther e exists a c ol le ction Q of p airwise vertex-disjoint squar e d-tight-p aths in H with V ( Q ) = V ( P ) ∪ L and a bije ction f : P → Q such that for e ach P ∈ P the p aths P and f ( P ) have the same ends. Pr o of. Apply Lemma 6.2 to obtain a a family F of at most β n pairwise-disjoint absorb ers such that for eac h v ∈ V ( H ) 4 there are at least β 2 n absorb ers for v in F , and so that the set V ( F ) of vertices co vered by F has | V ( F ) | ≤ 36 β n . The latter b ound implies that δ 2 ( H \ V ( F )) ≥ (7 / 9 + α/ 2) n , so H \ V ( F ) has at least 1 3 ·  n − 40 β n 2  ·  7 9 + α 2  n ≥ 7 9  n 3  edges. By Theorem 2.2 , w e hav e γ ( K 3 5 (4)) = γ ( K 3 5 ) ≤ 0 . 74. So b y Theorem 2.1 there m ust exist a cop y of K 3 5 (4) in H \ F , say with vertex classes { a k b k c k d k } for each k ∈ [5]. Set ˆ P := a 1 . . . a 5 b 1 . . . b 5 c 1 . . . c 5 d 1 . . . d 5 and P := { ˆ P } ∪ {P A : A ∈ F } . Note that |P | = 5 |F | + 1 ≤ 5 β n + 1 and | V ( P ) | ≤ | V ( F ) | +   V  K 3 5 (4)    ≤ 36 β n + 20 ≤ 40 β n . So it remains only to sho w that P has the claimed absorbing prop erty . Consider any set L ⊆ V ( H ) \ V ( P ) with | L | ≤ β 2 n . By p ossibly adding vertices from { a 5 , b 5 , c 5 } to L , form a set L ′ with L ⊆ L ′ ⊆ L ∪ { a 5 , b 5 , c 5 } such that | L ′ | is divisible by four. Set f ( ˆ P ) = ˆ P \ L ′ , and observe that f ( ˆ P ) is then a squared-tigh t-path w ith the same ends as ˆ P . Next, arbitrarily partition L ′ in to v ertex-disjoint ordered 4-tuples of distinct vertices v 1 , . . . , v t ∈ V ( H ) 4 , and write L = { v 1 , . . . , v t } . The num b er of these 4-tuples is |L| = | L ′ | / 4 ≤ ( β 2 n + 3) / 4 ≤ β 2 n , and each has at least β 2 n absorb ers in F , so we ma y greedily choose for eac h v ∈ L a unique A v ∈ F whic h is an absorb er for v . Let F L := { A v : v ∈ L} and let F c L := F \ F L . F or each A ∈ F L set f ( P ) = f A, v ( P ) for eac h P ∈ P A , and for each A ∈ F c L instead set f ( P ) = P for eac h P ∈ P A . So for each P ∈ P the image f ( P ) is then a squared-tight-path with the same ends as P . Set Q := { f ( P ) : P ∈ P } , and observ e that V ( Q ) = V ( P ) ∪ L , as desired. Finally , w e prov e Lemma 1.6 by connecting the vertex-disjoin t squared-tigh t-paths obtained from Lemma 6.3 into a single squared-tight-path. Pr o of of L emma 1.6 . Let P b e a collection of vertex-disjoin t squared-tight-paths with the prop erties stated in Lemma 6.3 , and write P = { P 1 , . . . , P N } . So in particular N ≤ 5 β n + 1 and | V ( P ) | ≤ 40 β n . F or each i ∈ [ N ], let y i b e the initial triple of P i and let x i b e the final triple of P i . Apply Lemma 1.5 with 40 β , N − 1 , y i +1 for each i ∈ [ N − 1] pla ying the roles of ψ , s, y i for each i ∈ [ s ] resp ectiv ely . This gives a collection P ′ = { P ′ 1 , . . . , P ′ N − 1 } of vertex-disjoin t squared-tight-paths in H \ X suc h that for each i ∈ [ N − 1] the path P ′ i has initial triple x i and final triple y i +1 , and also so that | V ( P ′ ) | ≤ √ 40 β n . Set P = P 1 P ′ 1 P 2 P ′ 2 . . . P N − 1 P ′ N − 1 P N , so | V ( P ) | ≤ 40 β n + √ 40 β n ≤ 7 √ β n . No w consider an y set L of at most β 2 n v ertices of V ( H ) \ V ( P ). By the absorbing prop ert y of P there exists a collection Q of pairwise v ertex-disjoin t squared-tigh t-paths in H with V ( Q ) = V ( P ) ∪ L and a bijection f : P → Q such that for each P ∈ P the paths P and f ( P ) hav e the same ends. F or eac h i ∈ [ N ] set Q i = f ( P i ), and set P ′ = Q 1 P ′ 1 Q 2 P ′ 2 . . . Q N − 1 P ′ N − 1 Q N . Then P ′ is a squared-tight- path in H with V ( P ′ ) = V ( P ) ∪ L and such that P and P ′ ha ve the same ends. 24 7 P ath Co v er Lemma In this section, w e pro ve the Path Cov er Lemma (Lemma 1.7 ). F or this we make use of the follo wing w eakened version of a theorem by Keev ash and Mycroft [ 14 ]. F or constants 1 /n ≪ 1 /ℓ ≪ γ ≪ β ≪ 1 /k , the full statement of the result assumes instead the w eaker edge-containmen t condition that for eac h i ∈ [ k − 1], each e ∈ J ( i ) is con tained in at least ( k − i k − γ ) n edges of J ( i +1) . The conclusion is then that either J ( k ) has a matc hing cov ering all but at most ℓ vertices, or that there is a subset S of V ( J ) with | S | = ⌊ j n/k ⌋ for some j ∈ [ k − 1] for which at most β n k edges of J ( k ) con tain more than j v ertices of S . The latter outcome is referred to as a ‘space barrier’, but a straightf orward coun ting argument sho ws this cannot o ccur under the stronger edge-con tainment assumption of the v ersion of the theorem stated b elo w. Theorem 7.1 ([ 14 , Theorem 2.4]) . L et 1 /n ≪ 1 /ℓ ≪ α ≪ 1 /k and let J b e a k -c omplex on n vertic es with J (1) = {{ v } : v ∈ V ( J ) } . If for e ach i ∈ [ k − 1] e ach e dge e ∈ J ( i ) is c ontaine d in at le ast ( k − i k + α ) n e dges of J ( i +1) , then J ( k ) c ontains a matching that c overs al l but at most ℓ vertic es of J . W e will apply Theorem 7.1 with k = 4 in the pro of of the follo wing lemma, sho wing that ev ery 3-graph H on n v ertices with minimum co degree at least 3 n/ 4 + o ( n ) admits an almost-spanning collection of pairwise vertex-disjoin t squared-tight-paths. Lemma 7.2 (Path Tiling Lemma) . L et 1 /n ≪ 1 /L ≪ γ , α ≤ 1 , and let H b e a 3 -gr aph on n vertic es with δ 2 ( H ) ≥ (3 / 4 + α ) n . Then ther e exists a c ol le ction of at most L p airwise vertex-disjoint squar e d-tight-p aths in H which c ol le ctively c over al l but at most γ n vertic es of H . Pr o of. Introduce new constants with 1 /n ≪ 1 /L ≪ 1 /t 1 ≪ 1 /t 0 ≪ 1 /T , 1 /r, ε ≪ c ≪ ε 3 , d 2 ≪ d 3 ≪ β ≪ θ ≪ γ , α. Arbitrarily delete at most t 1 ! vertices from H to obtain a subgraph H ′ on n ′ v ertices where t 1 ! divides n ′ . W e then hav e δ 2 ( H ′ ) ≥ (3 / 4 + α ) n − t 1 ! ≥ (3 / 4 + 5 α/ 6) n ′ . Apply Lemma 4.4 , with H ′ , n ′ and 3 pla ying the roles of G, n and k resp ectiv ely , to obtain a ( · , · , ε, ε 3 , r )-regular slice J for H ′ with clusters V 1 , . . . , V t , where t 0 ≤ t ≤ t 1 , and with density parameter d 2 . Let R := R d 3 ( H ′ ); recall that V ( R ) = [ t ] and that vertex i ∈ [ t ] corresp onds to the cluster V i of J . Let m be the num b er of v ertices in each cluster, so mt = n ′ . Prop osition 4.9 , with H ′ , 5 α/ 6 , 3 and θ playing the roles of H , α, γ and β resp ectiv ely , implies that R is (3 / 4 + 5 α / 12 , θ )-dense. By Lemma 4.10 , with R, t and 3 / 4 + 5 α/ 12 playing the roles of H , n and µ resp ectiv ely , there is a subgraph R ′ of R that is strongly (3 / 4 + α / 3 , 2 θ 1 / 4 )-dense, and which also satisfies | E ( R − R ′ ) | ≤ 8 √ θ  t 3  . Let U = { i ∈ V ( R ′ ) : deg ∂ R ′ ( i ) < (1 − θ 1 / 8 ) t } . Since | ∂ R ′ | ≥ (1 − 2 θ 1 / 4 )  t 2  , we hav e | U | ≤ 2 θ 1 / 8 t . Let R ∗ = R ′ \ U and t ′ = | V ( R ∗ ) | ≥ (1 − 2 θ 1 / 8 ) t . F or any pair of vertices uv ∈  V ( R ∗ ) 2  suc h that deg R ∗ ( uv ) > 0, we ha ve deg R ∗ ( uv ) ≥ deg R ′ ( uv ) − | U | ≥ (3 / 4 + α/ 3) t − 2 θ 1 / 8 t ≥ (3 / 4 + α/ 4) t ′ . Note that      V ( R ∗ ) 2  \ ∂ R ∗     ≤      V ( R ′ ) 2  \ ∂ R ′     ≤ 2 θ 1 / 4  t 2  ≤ 4 θ 1 / 4  t ′ 2  . By the ab ov e, we deduce that R ∗ is strongly (3 / 4 + α / 4 , 4 θ 1 / 4 )-dense and we ha ve δ ( ∂ R ∗ ) ≥ (1 − 4 θ 1 / 4 ) t ′ ≥ (3 / 4 + ε ) t ′ . (7.1) Define a 4-complex C with V ( C ) = V ( R ∗ ) in which C (1) = {{ i } : i ∈ V ( R ∗ ) } , C (2) = ∂ R ∗ , C (3) = E ( R ∗ ) , C (4) = E ( T ( R ∗ )) . 25 Claim 7.3. F or e ach i ∈ [3] e ach e dge e ∈ C ( i ) is c ontaine d in at le ast ( 4 − i 4 + ε ) t ′ e dges of C ( i +1) . Pr o of of claim. for i = 1 the claimed prop ert y holds b y ( 7.1 ). The fact that R ∗ is strongly dense implies that each ij ∈ ∂ R ∗ is contained in at least (3 / 4 + α/ 4) t ′ edges of R ∗ . This giv es the claimed prop ert y for i = 2, and also implies that for each edge ij k ∈ C (3) , each pair of v ertices from ij k has at most t ′ − (3 / 4+ α/ 4) t ′ ≤ (1 / 4 − α/ 4) t ′ non-neigh b ours in V ( C ). It follo ws that the n umber of options to c ho ose a v ertex ℓ such that ij k ℓ ∈ E ( T ( R ∗ )) is at least t ′ − 3(1 / 4 − α/ 4) t ′ = (1 / 4+ 3 α/ 4) t ′ ≥ (1 / 4+ ε ) t ′ , giving the claimed prop ert y for i = 3 also.  By Theorem 7.1 , with C , ε and T playing the roles of J , α and ℓ resp ectiv ely , the 4-graph C (4) con tains a matching cov ering all but at most T v ertices of C . This is a K 3 4 -tiling K in R cov ering all but at most 2 θ 1 / 8 t + T ≤ 3 θ 1 / 8 t vertices of R . Claim 7.4. L et i 1 , i 2 , i 3 , i 4 b e the vertic es of a c opy of K 3 4 in R . Then ther e is a c ol le ction P of at most 1 /c p airwise vertex-disjoint squar e d-tight-p aths in H [ S j ∈ [4] V i j ] which c overs al l but at most β m vertic es of e ach of V i 1 , V i 2 , V i 3 and V i 4 . Pr o of of claim. Let G (3) b e a squared-tight-path v 1 v 2 ...v 4 cm − 1 v 4 cm where v k ∈ V i j for all k ∈ [4 cm ] suc h that k ≡ j (mo d 4). Let G b e the do wn-closure of G (3) , so G is a 4-partite 3-complex with v ertex classes X 1 , . . . , X 4 where X j ⊆ V i j and | X j | = cm for all j ∈ [4]. Note that an y v ertex in G is in at most 9 edges in G (3) , therefore ∆ = ∆( G ) = 9 + 6 + 1 = 16. Let H b e the 4-partite 3-complex obtained from J [ S j ∈ [4] V i j ] by adding the edges of H ′ supp orted on J [ S ( j,k,ℓ ) ∈ ( [4] 3 ) V i j V i k V i ℓ ]. Therefore H is a ( d , ε k , ε, r )-regular 4-partite 3-complex with v ertex classes V i 1 , . . . , V i 4 all of size m , resp ecting the partition of G . Let P b e a maximal collection of v ertex-disjoint squared-tight-paths in H [ S j ∈ [4] V i j ], eac h of whic h uses exactly cm v ertices from eac h of V i 1 , . . . , V i 4 . In particular, the latter implies that |P | < 1 /c . Let V ′ i j := V i j \ V ( P ) for eac h j ∈ [4]. If |P | < (1 − β ) /c , then w e ha ve | V ′ i j | ≥ β m for each j ∈ [4]. Lemma 4.1 , with H , 3 , 4 , β , { V i j } j ∈ [4] pla ying the roles of G , k , s, α, { V i } i ∈ [ s ] resp ectiv ely , then implies that the induced sub complex H [ V ′ i 1 , . . . , V ′ i 4 ] is ( d , √ ε 3 , √ ε, r )-regular. So we ma y apply Lemma 4.8 , with H [ V ′ i 1 , . . . , V ′ i 4 ], 3, c/β , β m , √ ε 3 and √ ε pla ying the role of H , k , c, m, ε k and ε respectively , to obtain a partition-resp ecting copy of G in H [ V ′ i 1 , . . . , V ′ i 4 ]. This gives us a squared-tigh t-path P in H [ S j ∈ [4] V i j ] which uses exactly cm vertices from eac h of V i 1 , . . . , V i 4 , so the collection P ∪ P con tradicts the maximality of P . Therefore, we must hav e (1 − β ) /c ≤ |P | . It follo ws that for eac h j ∈ [4] we hav e | V i j \ V ( P ) | ≤ m − |P | cm ≤ β m . This pro ves the claim.  Applying Claim 7.4 to eac h member of K we obtain a collection of at most | K | /c ≤ t 1 / 4 c ≤ L v ertex-disjoint squared-tight-paths in H ′ whic h collectiv ely co ver all but at most β mt + 3 θ 1 / 8 mt + t 1 ! ≤ γ n vertices of H . W e are now ready to prov e Lemma 1.7 . W e do this b y connecting the squared-tigh t-paths obtained from Lemma 7.2 and the sp ecified ends e 1 and e 2 in to a single path. Pr o of of L emma 1.7 . Let e 1 and e 2 b e disjoint ordered triples whic h are edges of H . Let H ′ := H \ ( V ( e 1 ) ∪ V ( e 2 )) and n ′ := | V ( H ′ ) | . Note that δ 2 ( H ′ ) ≥ (7 / 9 + α ) n − 6 ≥ (7 / 9 + 3 α/ 4) n ′ . Cho ose a set W ⊆ V ( H ′ ) of size | W | = γ n ′ / 2 uniformly at random. Observ e that for eac h pair u, v of distinct v ertices of H , b oth deg H ( uv , W ) and deg H ( uv , V ( H ′ ) \ W ) are hypergeometric random v ariables, with exp ectations E deg H ( uv , W ) = (7 / 9 + 3 α/ 4) γ n ′ / 2 and E deg H ( uv , V ( H ′ ) \ W ) = (7 / 9 + 3 α/ 4)(1 − γ / 2) n ′ . So b y applying Ho effding’s inequalit y (Lemma 2.5 ) and taking a union 26 b ound ov er all pairs of vertices, we find that with high probabilit y every pair u, v of distinct vertices of H satisfies deg H ( uv , W ) ≥  7 9 + α 2  | W | and deg H  uv , V ( H ′ ) \ W  ≥  7 9 + α 2  | V ( H ′ ) \ W | . (7.2) Fix W for which this even t holds. Let n ′′ := | V ( H ′ ) \ W | , so δ 2 ( H ′ \ W ) ≥ (7 / 9 + α/ 2) n ′′ . By Lemma 7.2 , with H ′ \ W, n ′′ , α/ 2 and γ / 2 playing the roles of H, n, α and γ respectively , there exists a collection of at most L squared-tigh t-paths P 1 , . . . , P ℓ in H ′ \ W whic h collectiv ely co v er all but at most γ n ′′ / 2 v ertices of H ′ \ W . W e no w connect these squared-tigh t-paths using W . F or eac h i ∈ [ ℓ ] let p i b e the initial triple of P i and let q i b e the final triple of P i . Also let P 0 := p 0 = q 0 = e 1 and P ℓ +1 := p ℓ +1 = q ℓ +1 = e 2 . Claim 7.5. F or e ach 0 ≤ i ≤ ℓ , ther e exist disjoint or der e d triples x i and y i whose vertic es ar e in W such that (i) x i and y i ar e e dges of H [ W ] , (ii) x i p i and q i y i ar e b oth squar e d-tight-p aths, and (iii) the triples x i and y i for e ach 0 ≤ i ≤ ℓ ar e al l p airwise vertex-disjoint. Pr o of of claim. W e write p i = p i p ′ i p ′′ i and q i similarly . F or each 0 ≤ i ≤ ℓ we will pick v ertices x i , x ′ i , x ′′ i in order and write x i = x i x ′ i x ′′ i and similarly for y i . T o see that it is p ossible to choose such triples, for each 0 ≤ i ≤ ℓ in turn first choose x ′′ i in N H ( p i p ′ i , W ) ∩ N H ( p i p ′′ i , W ) ∩ N H ( p ′ i p ′′ i , W ), then choose x ′ i in N H ( p i p ′ i , W ) ∩ N H ( p i x ′′ i , W ) ∩ N H ( p ′ i x ′′ i , W ), then choose x i in N H ( p i x ′ i , W ) ∩ N H ( p i x ′′ i , W ) ∩ N H ( x ′ i x ′′ i , W ), and similarly choose y i , y ′ i , y ′′ i in that order. F or each choice, ( 7.2 ) ensures that the relev ant intersection of neighbourho o ds has size at least 3(7 / 9 + α/ 2) | W | − 2 | W | ≥ | W | / 3 ≥ 6( ℓ + 1), so there is alw ays an av ailable v ertex to choose whic h has not previously b een used.  Ha ving pick ed x i and y i for each 0 ≤ i ≤ ℓ , we now apply Lemma 1.5 , with W , 1 /L, θ n ′ , ℓ + 1 , y i and x i +1 for each 0 ≤ i ≤ ℓ playing the roles of H , ψ , n, s, x i and y i for each i ∈ [ s ] resp ectiv ely , to obtain a set of vertex-disjoin t squared-tigh t-paths P ′ 0 , . . . , P ′ ℓ suc h that for eac h 0 ≤ i ≤ ℓ , the path P ′ i has initial triple y i and final triple x i +1 . It follows that P = P 0 P ′ 0 P 1 P ′ 1 . . . P ′ ℓ P ℓ +1 is a squared-tigh t-path from P 0 = e 1 to P ℓ +1 = e 2 co vering all but at most γ n ′′ / 2 + | W | ≤ γ n v ertices of H . References [1] P . Allen, J. B¨ ottc her, O. Co oley, and R. Mycroft. “Tight cycles and regular slices in dense hypergraphs”. J. Combin. The ory Ser. 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