Ramsey theory of low-degree semialgebraic relations

Ramsey theory of lo w-degree semialgebraic relations Azem Adib elli ∗ and Istv´ an T omon ∗ Abstract W e pro ve that hypergraphs defined by lo w-degree p olynomial inequalities contain large homogeneous subsets. F ormally , let H b e an r -uniform h yp ergraph on N vertices that is semialgebraic of constan t description complexity , and eac h defining p olynomial has degree at most D . Then H con tains a clique or an independent set of size n , where N ≤ t w 3 D 3 ( n ). 1 In tro duction Ramsey’s theorem is a fundamental result in combinatorics and logic, stating that for every n there exists a smallest num b er N = R r ( n ) such that ev ery r -uniform N -v ertex hypergraph con tains a clique or an indep endent set of size n . It is a central topic in extremal combinatorics to understand the gro wth rate of R r ( n ), and to study ho w this function b ehav es in restricted h yp ergraph classes. The original pro of of Ramsey [ 20 ] gav e extremely p o or b ounds on R r ( n ). But so on after, Erd˝ os and Szek eres [ 15 ] and Erd˝ os [ 12 ] determined the righ t order in the case of graphs: R 2 ( n ) = 2 Θ( n ) . Finding the optimal constan t factor is a notorious open problem, with several exciting recent developmen ts [ 6 , 18 ], see the survey of Morris [ 19 ] for further discussion. F or r ≥ 3, Erd˝ os and Rado [ 14 ] and Erd˝ os, Ha jnal, and Rado [ 13 ] pro ved that t w r − 1 (Ω( n 2 )) < R r ( n ) < t w r ( O ( n )) , where the to wer function t w k ( n ) is defined recursiv ely by t w 1 ( x ) := x and t w k ( x ) := 2 tw k − 1 ( x ) . Determining the correct heigh t of the to w er remains op en, but m ulticolor v ersions of the problem indicate that the upper bound migh t be closer to the truth. F or more recent dev elopments on h yp ergraph Ramsey n umbers, we refer the reader to [ 10 ]. These results pain t a clear picture: while w e are guaranteed homogeneous sets (i.e., clique or indep enden t set) of increasing size, there are h yp ergraphs that only contain extremely small suc h sets. This limits the applications of general Ramsey results, where one is in terested in the size of homogeneous subsets of highly structured h yp ergraphs. Typical examples of such applications arise in geometry . F or instance, the Erd˝ os-Szekeres theorem [ 15 ] states that every sequence of N real n umbers a 1 , . . . , a N con tains a monotone subsequence of length at least √ N . Observ e that a w eaker bound of the form Ω(log N ) follows from the result R 2 ( n ) = 2 Θ( n ) : consider the graph G mon on vertex set [ N ] in whic h i < j are joined by an edge if a i < a j , and note that a homogeneous subset in this graph is a monotone subsequence. How ever, the sp ecial structure of this graph ensures muc h larger homogeneous sets. Another classical example is the Happ y Ending problem [ 15 ], asking for the smallest num b er N = K ( n ) such that an y set of N p oints in the plane contains n p oints in conv ex p osition. It is a simple exercise to show that K ( n ) ≤ R 3 ( n ), which follows by considering the N -v ertex 3-uniform hypergraph H cup , whose edges corresp ond to so called cups . This gives a double exp onential upp er b ound on K ( n ), while it is no w known [ 21 ] that K ( n ) = 2 n + o ( n ) . This motiv ates the question: for which natural h yp ergraph families can the tow er-type b ound R r ( n ) ≤ t w r ( O ( n )) b e significantly improv ed? Semialgebr aic hyp er gr aphs provide a unifying framework capable ∗ Ume ˚ a Univ ersity , e-mail : azem.adib elli@umu.se, istv antomon@gmail.com 1 of capturing a broad sp ectrum of geometrically defined structures. W e define an r -uniform hypergraph H to be semialgebraic of description complexity ( d, D , m ), if its v ertices are p oints in R d and its edge relation is determined b y a Bo olean com bination of m p olynomial inequalities of degree at most D (a formal definition is presen ted in the Preliminaries). F or example, the previously discussed graph G mon is semialgebraic of complexity (1 , 1 , 1), while H cup is semialgebraic of complexity (2 , 2 , 1). Under the assumption of fixed description complexity , man y problems that are hard for general h yp ergraphs b ecome more tractable. Let R d,D,m r ( n ) denote the smallest N suc h that an y r -uniform N -vertex semialgebraic h yp ergraph of complexity ( d, D , m ) con tains a clique or an independent set of size n . Clearly , R d,D,m r ( n ) ≤ R r ( n ). In the case of graphs, the function R d,D,m 2 ( n ) was first studied in the foundational pap er of Alon, P ach, Pinc hasi, Radoi ˇ ci ´ c and Sharir [ 2 ], where they prov ed that R d,D,m 2 ( n ) ≤ n O (1) . Here and later, the constant hidden by O ( . ) and Ω( . ) migh t dep end on r , d, D , m , but no other parameter. Recently , sharp b ounds on the exp onent were pro ved by Tidor and Y u [ 22 ]. Note that these represen t an exp onential improv emen t compared to the trivial b ound R d,D,m 2 ≤ R 2 ( n ) = 2 Θ( n ) . By adapting the inductiv e approach of Erd˝ os and Rado [ 14 ], Conlon, F o x, P ach , Sudak ov, and Suk [ 9 ] demonstrated that this improv ement extends to the h yp ergraph case as well: R d,D,m r ( n ) ≤ t w r − 1 ( n O (1) ) . More strikingly , they also show ed that this b ound is tigh t: for every uniformit y r ≥ 2, there exist parameters ( d, D , m ) suc h that R d,D,m r ( n ) ≥ t w r − 1 (Ω( n )). Consequen tly , the impro vemen t pro vided b y semialgebraic structure is only a single exp onential, lea ving the fundamental tow er-type dep endence caused by the uniformity r intact. A natural direction to circumv ent this is to consider how the sp ecific parameters of the description complexit y influence the growth rate of R d,D,m r ( n ). A b eautiful result in this direction, due to Bukh and Matou ˇ sek [ 5 ], establishes that R 1 ,D,m r ( n ) ≤ 2 2 O ( n ) for some c = c ( r , D , m ). That is, N -vertex semialgebraic h yp ergraphs of constant complexit y defined o ver the 1-dimensional real s pace alw ays contain homogeneous sets of size Ω(log log N ), indep endent of the uniformity . This b ound is also tigh t [ 5 , 9 ]. It remains an intriguing op en problem whether similar phenomena holds for d ≥ 2; sp ecifically , whether there exists t = t ( d ) such that R d,D,m r ( n ) ≤ t w t ( O ( n )). T ow ard a low er b ound, Eli´ a ˇ s, Matou ˇ sek, Rold´ an-P ensado, and Safernov´ a [ 11 ] provided a construction sho wing R d,D,m d +3 ( n ) ≥ t w d +2 (Ω( n )). Another direction is to b ound the parameter D denoting the degree of defining p olynomials. In recen t y ears, hypergraphs defined b y linear inequalities, or equiv alently , semialgebraic h yp ergraphs of complexit y ( d, 1 , m ), ha ve received substantial in terest. This is largely due to a result of Basit, Chernik ov, Starc henko, T ao, and T ran [ 4 ], whic h demonstrated that such hypergraphs exhibit in teresting b eha vior with resp ect to Zarankiewicz-type problems, a topic which has since seen significant dev elopment [ 7 , 8 , 16 ]. The authors also coined the term semiline ar hyp er gr aph for this family . Ramsey-type problems for semilinear graphs w ere studied by T omon [ 23 ], while for semilinear hypergraphs, Jin and T omon [ 17 ] pro ved the optimal b ound R d, 1 ,m r ( n ) ≤ 2 n O (1) . Th us, restricting the degree to D = 1 has a similar effect on the Ramsey n umber as restricting the dimension to d = 1. The main result of our pap er sho ws that for any fixed degree D , the Ramsey n umber R d,D,m r ( n ) can b e b ounded by t w t ( O ( n )), where the tow er heigh t t = t ( D ) only dep ends on D . This provides a substan tial improv ement ov er the R d,D,m r ( n ) ≤ tw r − 1 ( O ( n )) b ound for hypergraphs defined b y low degree p olynomials. 2 Theorem 1.1. L et H b e an r -uniform semialgebr aic hyp er gr aph of description c omplexity ( d, D , m ) on N vertic es, wher e N ≫ d, D , m, r . Then H c ontains a clique or an indep endent set of size n , wher e N ≤ tw 3 D 3 ( n ) . In Subsection 5.3 , we deduce Theorem 1.1 from a more general result, whic h applies to a larger class of hypergraphs. The to wer height O ( D 3 ) in Theorem 1.1 is likely far from optimal, but it cannot b e replaced by a v alue smaller than D − 1. This is due to the fact that any r -uniform semialgebraic h yp ergraph of description complexit y ( d, D , m ) also has description complexit y ( d ′ , r , m ) for some d ′ dep ending only on r, d, D . This follo ws from a standard application of V eronese mappings. Since it is kno wn that there exist constructions achieving the b ound R d,D,m ( n ) ≥ t w r − 1 (Ω( n )), setting D = r also sho ws that R d ′ ,D,m D ( n ) ≥ t w D − 1 (Ω( n )) with suitable d ′ and m . Organization. In the next section, we presen t our notation and a few simple auxiliary results. In Section 3 , we give a rough outline of the pro of of Theorem 1.1 . Then, in Section 4 , we present the pro of of the D = 2 case, in order to illustrate our k ey new ideas in a less technical manner. The detailed pro of of Theorem 1.1 is then presen ted in Section 5 . 2 Preliminaries First, we in tro duce our terminology . W e use mostly standard graph and set theoretic notation. W e write [ n ] := { 1 , . . . , n } . F or ease of notation, we declare certain v ariables as constants, and then the O ( . ) and Ω( . ) notations hide factors that p ossibly dep end on these constants, but no other v ariables. Moreov er, if w e say a quantit y is sufficiently large, it is with resp ect to constants. The tower function tw k ( x ) is defined as t w 1 ( x ) := x and tw k ( x ) := 2 tw k − 1 ( x ) for k ≥ 2. W e highligh t the simple iden tity tw k (t w ℓ ( x )) = t w k + ℓ − 1 ( x ). An or der e d set refers to any set with a complete ordering on its elements. W e denote this ordering b y simply < . Given an ordered set A and in teger k , we write (unconv en tionally) A ( k ) := { ( a 1 , . . . , a k ) ∈ A k : a 1 < · · · < a k } . Also, write A ( ≤ k ) := [ 0 ≤ ℓ ≤ k A ( ℓ ) , where w e use the conv en tion that A (0) = { () } . Given a = ( a 1 , . . . , a k ) ∈ A k and I ⊂ [ k ], a I := (( a i ) i ∈ I ) . With sligh t abuse of notation, if a ∈ A k and b ∈ A m , w e write ( a, b ) for ( a 1 , . . . , a k , b 1 , . . . , b m ) ∈ A k + m 2.1 Semialgebraic hypergraphs W e giv e a formal definition of semialgebraic hypergraphs of description complexit y ( d, D, m ), where d refers to the dimension of the ambien t space, D is an upp er b ound on the (total) degree of the defining p olynomials, and m is an upp er b ound on the num b er of defining p olynomials. Definition 1 (Semialgebraic hypergraph) . Let V ⊂ R d b e an ordered set. Let f 1 , . . . , f m : ( R d ) r → R b e p olynomials of degree at most D , and let Γ : { false , true } m → { false , true } b e a Bo olean formula. Define the r -uniform h yp ergraph H on v ertex set V in which x ∈ V ( r ) forms an edge if Γ ( f 1 ( x ) ≤ 0 , . . . , f m ( x ) ≤ 0) = true . Then H , and any hypergraph isomorphic to H , is a semialgebraic hypergraph of description complexit y ( d, D , m ). 3 W e remark that sev eral closely related definitions of semialgebraic h yp ergraphs app ear in the literature. F or instance, Conlon, F ox, Pac h, Sudako v, and Suk [ 9 ] define complexit y as a pair ( d, t ), where t serv es as a simultaneous upp er b ound for b oth m and D . In con trast, Tidor and Y u [ 22 ] defines complexit y ( d, D ), where D is the sum of the degrees of the m p olynomials f 1 , . . . , f m . Moreov er, they w ork with unordered vertex sets V , but in exchange the edge relation has to b e assumed symmetric. Our c hoice of ( d, D , m ) provides a finer description of the complexit y , as we are mainly interested in the degrees D . How ever, our definition do es not c hange the core concept of semialgebraic relations. 2.2 Exp onen tial sequences A core comp onen t of the pro of of our main theorem is the study of exp onential subsequences, which we discuss in detail in this section. Definition 2 (Exp onential sequence) . A sequence of real num b ers { a i } i =1 ,...,n is • exp onen tial if 0 ≤ 2 a i − 1 ≤ a i for i = 2 , . . . , n ; • w eakly-exp onential of type T1 if { a i } i =1 ,...,n is exp onen tial, T2 if { a i } i = n,..., 1 is exp onen tial, T3 if {− a i } i =1 ,...,n is exp onen tial, T4 if {− a i } i = n,..., 1 is exp onen tial; • shifted-exp onen tial of type τ ∈ { T1 , T2 , T3 , T4 } if there exists t ∈ R such that { a i − t } i =1 ,...,n is w eakly-exp onential of type τ . The strength of shifted-exp onen tial sequences comes from the fact that any sequence of n real n umbers con tains a shifted-exp onential subsequence of length Ω(log n ), and this b ound is tight [ 17 ]. This can b e view ed as an analogue of the celebrated Erd˝ os-Szekeres theorem [ 15 ], whic h ensures that any sequence of length n contains a monotone subsequence of length ⌈ √ n ⌉ . An important idea, whic h also play ed a k ey role in the work of Jin and T omon [ 17 ], is that shifted-exp onential subsequences can b e found with the help of coloring triplets. This motiv ates the following definition. Definition 3. A triple of real num b ers ( a, b, c ) has type T1 if 0 ≤ b − a ≤ c − b , T2 if b − a ≥ c − b ≥ 0, T3 if 0 ≤ a − b ≤ b − c , T4 if a − b ≥ b − c ≥ 0, T0 if a, b, c is not monotone. In case a triple or sequence has more than one t yp e, we assign a type arbitrarily . Lemma 2.1. L et { a i } i =1 ,...,n b e a se quenc e of r e al numb ers. If every triple ( a i , a j , a k ) for 1 ≤ i < j < k ≤ n has typ e τ ∈ { T1 , T2 , T3 , T4 } , then { a i } i =1 ,...,n is shifte d-exp onential of typ e τ . Mor e over, if every triple has typ e T0 , then n ≤ 4 . 4 Pr o of. It is an easy exercise to show that an y sequence of 5 num b ers con tains a subsequence of size 3 that is monotone, so a sequence can only b e t yp e T0 if it has at most 4 elements. No w assume that { a i } i =1 ,...,n has a non- T0 type. W e only consider t yp e T1 , the other three cases can b e handled similarly . W e show that { a i − a 1 } i =1 ,...,n is exp onen tial, which then implies that { a i } i =1 ,...,n is shifted-exp onen tial of type T1 . Indeed, ( a 1 , a i , a i +1 ) is of t yp e T1 , which means that 0 ≤ a i − a 1 ≤ a i +1 − a i , whic h then implies 0 ≤ 2( a i − a 1 ) ≤ a i +1 − a 1 . In general, given a function ϕ : A → R (or sequence of real num b ers), a shifting of this function refers to any function of the form ϕ − t for some t ∈ R . W e also define a relaxation of exp onential sequences to general real functions. This notion will b e used in the case of functions acting on D -tuples, i.e. ϕ : A ( D ) → R . Definition 4. A function ϕ : B → R is exp onentially separated if ϕ ( b ) has the same sign for every b ∈ B , and for any tw o distinct b  = b ′ , w e hav e | ϕ ( b ) | ≥ 2 | ϕ ( b ′ ) | or | ϕ ( b ′ ) | ≥ 2 | ϕ ( b ) | . W e define exp onentially separated sequences analogously . Equiv alently , ϕ : B → R is exp onentially separated if and only if there is an enumeration b 1 , . . . , b n of the elemen ts of B suc h that { ϕ ( b i ) } i =1 ,...,n is exp onen tial (or weakly-exponential). 2.3 Extremal combinatorics W e presen t some simple or well known auxiliary results ab out h yp ergraphs. First, we state the quan titative v ersion of Ramsey’s theorem for colorings with constant n umber of colors, pro ved originally b y Erd˝ os and Rado [ 14 ]. Theorem 2.2. L et r , s b e p ositive inte ger c onstants. L et H b e the c omplete r -uniform hyp er gr aph on N vertic es, whose e dges ar e c olor e d with s c olors. Then H c ontains a mono chr omatic clique of size n , wher e N ≤ t w r ( O ( n )) . W e also mak e use of the follo wing lemma, which can b e thought of as an extension of the so called an ti-Ramsey theorem of Babai [ 3 ], see also [ 1 ]. Lemma 2.3. L et r , s, q b e p ositive inte ger c onstants. L et H b e the c omplete r -uniform hyp er gr aph on N vertic es, and let γ : E ( H ) → R s satisfy the fol lowing pr op erty. F or every e dge f , ther e ar e at most q e dges f ′ such that | f ∩ f ′ | = r − 1 and | γ ( f ) i − γ ( f ′ ) i | < 1 for some i ∈ [ s ] . Then ther e exists a c omplete subhyp er gr aph H ′ on Ω( N 1 / (2 r − 1) ) vertic es such that | γ ( f ) i − γ ( f ′ ) i | ≥ 1 for every distinct p air of e dges f , f ′ ∈ E ( H ′ ) and i ∈ [ s ] . Pr o of. Let S = { ( f , f ′ ) ∈ E ( H ) : f  = f ′ , ∃ i ∈ [ s ] , | γ ( f ) i − γ ( f ′ ) i | < 1 } , so that S is the set of ”conflicts”. Fix f ∈ E ( H ), then # { f ′ : ( f , f ′ ) ∈ S, | f ∩ f ′ | = k } ≤ 2 r q sN r − k − 1 . Indeed, for each g ⊂ f of size k , the num b er of f ′ suc h that ( f , f ′ ) ∈ S and g = f ∩ f ′ is at most q sN r − k − 1 . This follo ws from the observ ation that the n umber of ( r − 1)-tuples g is con tained in is at most N r − k − 1 , and eac h ( r − 1)-tuple can b e contained in at most q s such edges f ′ . 5 The rest of the pro of is a standard application of the probabilistic deletion method. Let p = cN − 2 r − 2 2 r − 1 with c = 1 / (2 r +1 r q s ), and sample the v ertices of H with probabilit y p , with U denoting the set of sampled v ertices. Let X b e the num b er of pairs ( f , f ′ ) ∈ S suc h that f , f ′ ⊂ U . Then E ( X ) = X ( f ,f ′ ) ∈ S p | f ∪ f ′ | = r − 1 X k =0 p 2 r − k · # { ( f , f ′ ) ∈ S : | f ∩ f ′ | = k } ≤ r − 1 X k =0 p 2 r − k N r · (2 r q sN r − k − 1 ) = 2 r q s r − 1 X k =0 c 2 r − k N 2 r − k 2 r − 1 − 1 < 2 r r q sc r +1 N 1 2 r − 1 < c 2 N 1 2 r − 1 . F or eac h ( f , f ′ ) ∈ S such that f , f ′ ⊂ U , delete a v ertex of f ∪ f ′ from U , and let U ′ b e the resulting set. Then E ( | U ′ | ) ≥ E ( | U | − X ) ≥ cN 1 / (2 r − 1) / 2. Fix an outcome of the sampling suc h that | U ′ | ≥ cN 1 / (2 r − 1) / 2, then H [ U ′ ] satisfies the required prop erties. 3 Pro of outline W e give a rough outline of the pro of of Theorem 1.1 . First, we discuss the pro of of Jin and T omon [ 17 ] of the D = 1 case, then we present our new k ey ideas by considering the case D = 2. F or further simplification, we assume that the num b er of defining p olynomials is m = 1, b y noting that the analysis of the general case do es not increase the complexit y significantly . First, let H b e an N -v ertex r -uniform semialgebraic h yp ergraph of description complexity ( d, 1 , 1). Then there exists a linear function f : ( R d ) r → R such that x ∈ V ( H ) ( r ) is an edge if and only if f ( x ) ≤ 0. Using that f is linear, we can write f ( x ) = P r i =1 ϕ i ( x i ) with suitable functions ϕ i : R d → R , i ∈ [ r ]. The k ey idea is to consider the sequences { ϕ i ( x i ) } x ∈ V ( H ) for i ∈ [ r ], and find a set S ⊂ V ( H ) of size (log N ) Ω(1) suc h that all r sequences { ϕ i ( x i ) } x ∈ S are shifted-exp onen tial. After shifting, w e get functions σ i : S → R and a constant C such that f ( x ) = C + P r i =1 σ i ( x i ), and each sequence { σ i ( x i ) } x ∈ S is w eakly-exp onen tial. The adv antage of this is that for a ”generic” r -tuple x ∈ S ( r ) , the sum C + P r i =1 σ i ( x i ) is dominated by the term with the largest absolute v alue, so its sign is determined b y the sign of this term. T o finish the pro of, w e find T ⊂ S of size | S | Ω(1) suc h that every x ∈ T ( r ) is generic, and the index of the largest element of ( C , | σ 1 ( x 1 ) | , . . . , | σ r ( x r ) | ) do es not dep end on the c hoice of x . This ensures that f ( x ) has the same sign for ev ery x ∈ T ( r ) , so T is either a clique or an indep enden t set of size (log N ) Ω(1) in H . Next, let H b e an N -v ertex r -uniform semialgebraic h yp ergraph of description complexity ( d, 2 , 1). Then there exists f : ( R d ) r → R of degree at most 2 such that x ∈ V ( H ) ( r ) is an edge if and only if f ( x ) ≤ 0. Using that f has degree at most 2, we can write f ( x ) = P I ∈ [ r ] (2) ϕ I ( x I ) with suitable functions ϕ I : ( R d ) 2 → R , I ∈ [ r ] (2) . No w ϕ I acts on the pair of vertices of V ( H ), so { ϕ I ( x I ) } x ∈ V ( H ) corresp onds to an edge w eighting of a complete graph instead of a sequence. Therefore, if one wan ts to adapt our approach for the D = 1 case, one needs an analogue of shifted-exp onential sequences for graph weigh tings. Our ultimate goal is to find a large subset S suc h that σ I : S (2) → R is exp onen tially separated for every I ⊂ [ r ] (2) , where σ I is a slightly mo dified v ersion of ϕ I . Unfortunately , allowing only shiftings will not b e sufficien t to achiev e this goal. Indeed, consider the following example. Let ϕ : [ n ] (2) → R b e defined as ϕ ( a, b ) = n a + b . It is easy to see that no shifting of ϕ is exp onen tially separated on more than 3 elements of [ n ]. The issue is that ϕ ( a, b ) ≈ ρ ( a ) for some function ρ only dep ending on a . T o ov ercome this, we allow functions σ of the form σ ( a, b ) = ϕ ( a, b ) − ρ 1 ( a ) − ρ 2 ( b ), and write ϕ ∼ σ for all such functions σ . This resolves the issue: w e sho w that there exist S ⊂ V ( H ) and σ I : S (2) → R , I ∈ [ r ] (2) , such that N ≤ tw 8 ( O ( | S | 3 )), ϕ I ∼ σ I 6 and σ I is exp onen tially separated. No w, we can rewrite f ( x ) = X I ∈ [ r ] (2) σ I ( x I ) + X i ∈ [ r ] ϕ i ( x i ) with suitable functions ϕ i : S → R . W e further pass to a subset T ⊂ S of size (log | S | ) Ω(1) suc h that { ϕ i ( x i ) } x ∈ T is shifted-exp onential for ev ery i ∈ [ r ], and rewrite f ( x ) = C + X I ∈ [ r ] (2) σ I ( x I ) + X i ∈ [ r ] σ i ( x i ) , (1) with suitable C ∈ R and σ i , i ∈ [ r ], eac h of which is w eakly-exp onential on T . The rest of the proof is very similar as in the D = 1 case. W e observe that for a ”generic” x ∈ T ( r ) , the sum in the righ t-hand-side of ( 1 ) is dominated by one of the terms, so the sign of f is the same as the sign of this dominant term. T o finish the pro of, we find U ⊂ T such that | T | ≤ tw 4 ( O ( | U | )), every x ∈ U ( r ) is generic, and the index of the largest element of ( C , ( | σ i ( x i ) | ) i ∈ [ r ] , ( σ I ( x I )) I ∈ [ r ] (2) ) do es not dep end on the c hoice of x . This ensures that f ( x ) has the same sign for every x ∈ U ( r ) , so U is either a clique or an indep endent set in H . The most difficult part of the pro of is finding the desired set S and functions σ I . F or simplicit y , consider a single function ϕ : A (2) → R on some finite ordered set A . W e define the ev aluation of ϕ as a function ψ ϕ acting on the 4-tuples of A suc h that for ( a, b, c, d ) ∈ A (4) , ψ ϕ ( a, b, c, d ) = ϕ ( a, c ) − ϕ ( a, d ) − ϕ ( b, c ) + ϕ ( b, d ) . Here, ψ ϕ has the adv antage that ψ ϕ ≡ ψ σ for ev ery ϕ ∼ σ . In tw o rounds, we define certain colorings on the 5- and then 4-tuples of S with the help of ψ ϕ , and use Ramsey’s theorem to find a large mono chromatic set B . This set B allo ws the construction of some σ ∼ ϕ that is exp onentially separated on B . 4 W arm-up: degree 2 semialgebraic h yp ergraphs In this section, w e pro ve the ( D , m ) = (2 , 1) sub case of T heorem 1.1 in a somewhat simplified manner. This serves to illustrate the k ey ideas of the general pro of while sidestepping the more technical details. Giv en an ordered set A , define an equiv alence relation on the space of functions { ϕ : A (2) → R } as follo ws. W rite ϕ ∼ ϕ ′ if there exist functions ρ 1 , ρ 2 : A → R such that ϕ ( a, b ) = ϕ ′ ( a, b ) − ρ 1 ( a ) − ρ 2 ( b ) for ev ery ( a, b ) ∈ A (2) . Giv en a function ϕ : A (2) → R , we define the ev aluation ψ ϕ : A (4) → R such that ψ ϕ ( a, b, c, d ) = ϕ ( a, c ) − ϕ ( a, d ) − ϕ ( b, c ) + ϕ ( b, d ) . The adv antage of ψ ϕ is that it is inv ariant on the equiv alence class of ϕ , that is, if ϕ ∼ ϕ ′ , then ψ ϕ ≡ ψ ϕ ′ . If ϕ is clear from the con text, w e write simply ψ instead of ψ ϕ . The follo wing is our main technical lemma. Lemma 4.1. L et ϕ : [ N ] (2) → R . Then ther e exist S ⊂ [ N ] of size n for some N ≤ t w 8 ( O ( n 3 )) and σ : S (2) → R such that σ ∼ ϕ on S and σ is exp onential ly sep ar ate d. Pr o of. Define a coloring of [ N ] (5) with a color from { T0 , . . . , T4 } . W e color ( a, b, c, d, e ) dep ending on the relation of X = ψ ( a, b, d, e ) and Y = ψ ( b, c, d, e ), where ψ = ψ ϕ . W e use color T1 if 0 ≤ X ≤ Y , 7 T2 if 0 ≤ Y ≤ X , T3 if 0 ≤ ( − X ) ≤ ( − Y ), T4 if 0 ≤ ( − Y ) ≤ ( − X ), T0 if X and Y has different signs. Using the qualitative form of Ramsey’s theorem (Theorem 2.2 ), there exists a mono c hromatic S 1 ⊂ [ N ] suc h that N ≤ tw 5 ( O ( | S 1 | )). Let τ 1 denote the color of S 1 . Assuming N is sufficiently large, w e ha ve τ 1  = T0 , as an y set completely colored with T0 has size at most 6. F or ease of notation, relab el the elemen ts of S 1 suc h that S 1 = [ N 1 ]. Fix some d ≤ N 1 − 1, and consider the sequence x ( d ) a = x a = ϕ ( a, d + 1) − ϕ ( a, d ) for a = 1 , . . . , d − 1. W e ha ve ψ ( a, b, d, d + 1) = x b − x a , so the fact that S 1 is mono c hromatic of color τ 1 implies that { x a } a =1 ,...,d − 1 is a shifted-exp onential sequence of type τ 1 b y Lemma 2.1 . Therefore, there exists t d suc h that { x ( d ) a − t d } a =1 ,...,d − 1 is w eakly-exp onential of type τ 1 . Let u d = P i 5, the color of S 2 is not type T0 . T o simplify notation, relab el the elemen ts of S 2 suc h that S 2 = [ N 2 ]. Then S 2 b eing mono chromatic implies that for every a ∈ [ N 2 ], the sequence { ϕ 1 ( a, b ) } b = a +1 ,...,N 2 is shifted-exp onential of type τ 2 for some τ 2 ∈ [4]. Therefore, there exists w a suc h that { ϕ 1 ( a, b ) − w a } b = a +1 ,...,N 2 is w eakly-exp onential of type τ 2 . Define σ : S (2) 2 → R such that σ ( a, b ) = ϕ 1 ( a, b ) − w a , and observe that σ ( a, e ) − σ ( a, d ) = ϕ 1 ( a, e ) − ϕ 1 ( a, d ) for ev ery a < d < e . F or conv enience, we collect the imp ortan t prop erties of σ : • σ ∼ ϕ 1 ∼ ϕ on S 2 , • { σ ( a, b ) } b = a +1 ,...,N 2 is w eakly-exp onential of type τ 2 for ev ery a ∈ S 2 , • { σ ( a, e ) − σ ( a, d ) } a =1 ,...,d − 1 is w eakly-exp onential of type τ 1 for ev ery d < e ≤ N 2 . 8 F or the sak e of simplicity , we assume that ( τ 1 , τ 2 ) = ( T1 , T1 ). The other cases can b e handled similarly , whic h we will demonstrate in the detailed pro of of the more general Lemma 5.2 . Let S 3 b e the subset of ev en num b ers in S 2 = [ N 2 ]. W e sho w that S 3 satisfies the follo wing prop erty: 4. { σ ( a, b ) } a ∈ S 3 : a 0 and X i ∈ [1 / 2 , 1] Y i , so X i > 0 as well. But this con tradicts τ k,m = T3 . All further ”Not p ossible” cases are due to similar reason. (T1,T4) Not p ossible. (T2,T1) Not p ossible. (T2,T2) Not p ossible. 16 (T2,T3) W e ha ve 0 ≤ 2 Z 1 ≤ ( − 4 X 1 ) ≤ ( − X 2 ) ≤ Z 2 , so n ψ ( k +1) ( a, b, c, d, e ) o b ∈ U ℓ +1 : a m − 1 4 r 2 D . But then there is p suc h that 2 r − D | σ f ,I ( a ( p ) J ) | ≥ | σ f ,I ′ ( a ( p ) J ′ ) | or 2 r D | σ f ,I ( a ( p ) J ) | ≤ | σ f ,I ′ ( a ( p ) J ′ ) | , con tradiction. Therefore, we ma y assume that T is mono chromatic of some color χ  = 0. Let f ∈ F and x ∈ T ( r ) , and consider f ( x ) = X I ∈ [ r ] ( ≤ D ) σ f ,I ( x I ) . Assume that I f ,x 0 = I 0 ∈ [ r ] ( ≤ D ) maximizes | σ f ,I ( x I ) | , then using that T is not of color 0, we hav e r 2 D | σ f ,I ( x I ) | ≤ | σ f ,I 0 ( x I 0 ) | for ev ery I  = I 0 . But as the sum contains | [ r ] ( ≤ D ) | < 2 r D elemen ts, this means that the sign of f ( x ) is the same as the sign of σ f ,I 0 ( x I 0 ). The k ey observ ation is that I 0 do es not dep end on the choice of x . Thi s follo ws from the more general statement that the order of the elements of the sequence ( σ f ,I ( x I )) I ∈ [ r ] ( ≤ D ) is completely determined b y the color χ . Indeed, let I , I ′ ∈ [ r ] ( ≤ D ) suc h that I  = I ′ , and c ho ose a (2 D )-tuple a ∈ T (2 D ) whose first | [ I ] ∪ [ I ′ ] | elemen ts are the elemen ts of [ I ] ∪ [ I ′ ]. Then σ f ,I ( x I ) = σ f ,I ( a J ) and σ f ,I ′ ( x I ′ ) = σ f ,I ′ ( a J ′ ) for some J  = J ′ ⊂ [2 D ] that dep end only on I and I ′ . As a is colored b y χ , the relative order of | σ f ,I ( a J ) | and | σ f ,I ′ ( a J ′ ) | is completely determined b y χ . T o summarize, w e ha v e shown that for every f ∈ F , there exists I f 0 ∈ [ r ] ( ≤ D ) suc h that the sign of f ( a ) is the same as the sign of σ f ,I 0 ( a I 0 ). Moreo ver, using that σ f ,I 0 is exponentially separated on T , this tells us that f ( a ) has the same sign for every choice of a . Thus, { f ( a ) ≤ 0 } f ∈ F ∈ { false , true } F is constan t on T , which means that H [ T ] is a clique or an indep endent set. W e finish the pro of by b ounding the size of T . W e hav e | S 0 | = N and | S k | ≤ tw K k ( | S k +1 | O (1) ) with K k = 3( D − k ) 2 / 2 + ( D − k ) / 2 + 1 for k = 0 , . . . , D − 1. Therefore, N ≤ t w L ( | S | O (1) ) with L = − ( D − 1) + D − 1 X k =0 3( D − k ) 2 2 + D − k 2 + 1 = D ( D + 1) 2 2 + 1 . Finally , we ha ve | S | ≤ t w 2 D ( O ( | T | )), which gives N ≤ t w K ( O ( | T | )) with K = D ( D 2 +6 D +1) 2 . W e observe that the b ound K = D ( D 2 +6 D +1) 2 satisfies K < 3 D 3 if D ≥ 2. F or D = 1, w e know the stronger b ound N ≤ 2 n O (1) from [ 17 ], so Theorem 1.1 holds for all v alues of D . 20 Ac kno wledgmen ts Both authors ac knowledge the supp ort of the Swedish Research Council gran t VR 2023-03375. W e would lik e to thank Zhihan Jin and J´ anos Pac h for v aluable discussions. References [1] N. Alon, T. Jiang, Z. Miller, and D. Pritikin. Pr op erly c olor e d sub gr aphs and r ainb ow sub gr aphs in e dge-c olorings with lo c al c onstr aints. Random Structures & Algorithms 23.4 (2003): 409–433. [2] N. 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