EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD

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1 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT Dedicated to Professor Jaroslav Nešetřil o the occasio of his 60th birthday Abstract. Szemerédi s regularity lemma asserts that every graph ca be decomposed ito relatively few radom-lie subgraphs. This radom-lie behavior eables oe to fid ad eumerate subgraphs of a give isomorphism type, yieldig the so-called coutig lemma for graphs. The combied applicatio of these two lemmas is ow as the regularity method for graphs ad has proved useful i graph theory, combiatorial geometry, combiatorial umber theory ad theoretical computer sciece. Recetly, the graph regularity method was exteded to hypergraphs by Gowers ad by Soa ad the authors. The hypergraph regularity method has bee successfully employed i a hadful of combiatorial applicatios, icludig alterative proofs to well-ow desity theorems of Szemerédi ad of Fursteberg ad Katzelso. I this paper, we apply the hypergraph regularity method to a few extremal hypergraph problems of Ramsey ad Turá flavor. 1. Itroductio Szemerédi s regularity lemma asserts that every graph ca be decomposed ito a bouded umber of so-called ε-regular pairs. For a graph G = (V, E) ad ε > 0, we say two o-empty disjoit subsets X, Y V are ε-regular if for all X X, X > ε X ad Y Y, Y > ε Y, we have d G (X, Y ) d G (X, Y ) < ε, where d G (X, Y ) = G[X, Y ] /( X Y ) is the desity of the bipartite subgraph G[X, Y ] of G (cosistig of all edges {x, y} E with x X ad y Y ). Szemerédi s lemma is the give as follows. Theorem 1.1 (Szemerédi s regularity lemma). For every ε > 0 ad iteger t 0, there exist itegers T 0 = T 0 (ε, t 0 ) ad N 0 = N 0 (ε, t 0 ) so that for every graph G = (V, E), V N 0, V admits a partitio V = V 1 V t, t 0 t T 0, satisfyig (i ) V 1 V t V ad (ii ) all but at most ε ( t 2) pairs (Vi, V j ), 1 i < j t, are ε-regular. Partitios V = V 1 V t satisfyig (i ) ad (ii ) as above are said to be t- equitable ad ε-regular. Szemerédi s regularity lemma lead to may applicatios i combiatorial mathematics, particularly i the area of extremal graph theory Date: February 1, Mathematics Subject Classificatio. Primary: 05C35. Secodary: 05C65, 05D10. Key words ad phrases. Turá s theorem, Ramsey theory, removal lemma, regularity lemma for hypergraphs. The first author was partially supported by NSF grat DMS The secod author was partially supported by NSF grat DMS The third author was supported by DFG grat SCHA 1263/1-1. 1

2 2 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT (see [19, 20] for surveys). May applicatios of Szemerédi s lemma deped o the fact that withi a appropriately give ε-regular partitio V = V 1 V t, oe may eumerate small subgraphs of a fixed isomorphism type. This result is formally due to the followig easily proved coutig lemma for graphs. I Fact 1.2 below ad elsewhere i this paper, we write x = y ± ξ for reals x ad y ad some positive ξ > 0 for the iequalities y ξ < x < y + ξ. Fact 1.2 (Graph coutig lemma). For all d > 0, γ > 0 ad every positive iteger l, there exist ε > 0 ad 0 so that wheever G is a l-partite graph with l- partitio V 1 V l, ad V 1 = = V l = 0, satisfyig for all 1 i < j l (a ) d G (V i, V j ) = d ± ε ad (b ) (V i, V j ) is ε-regular, the the umber K l (G) of l-cliques i G satisfies K l (G) = d (l 2) l (1 ± γ). We refer to a joit applicatio of Theorem 1.1 ad Fact 1.2 as the regularity method for graphs. Perhaps oe of the first applicatios of this method is due to Ruzsa ad Szemerédi [36] who showed it ca be used to prove Roth s theorem [34, 35], i.e., Theorem 1.4 below for d = 1 ad l = 3. More formally, Ruzsa ad Szemerédi used the graph regularity method to prove that every graph G o vertices havig o( 3 ) triagles cotais a triagle-free subgraph G havig oly o( 2 ) fewer edges. Their result ca be referred to as the triagle removal lemma (cf. Theorem 1.3 below) ad implies, as a corollary, Roth s theorem. I what follows, a hypergraph H 2 V with vertex set V is a collectio of subsets from V. We say H () is a -uiform hypergraph, or -graph, for short, if every subset belogig to H () has cardiality. A extesio of Szemerédi s regularity lemma for 3-graphs has bee developed i [10]. More recetly, extesios to -graphs were obtaied by Gowers [12, 13] ad by Soa ad the curret authors [22, 33], ad based o that wor, subsequetly by Tao [41] ad the secod two authors [29]. Usig these techiques, a hadful of 3-graph applicatios appear i [3, 10, 14, 17, 18, 21, 27, 28, 37, 38] ad some applicatios for -graphs appear i [22, 30, 31, 32] (some of which we discuss mometarily). Tao also obtaied some deep umber-theoretic applicatios i [40]. The goal of this paper is to use the hypergraph regularity method established i [22, 33] to ivestigate some extremal hypergraph problems (see Sectio 2). The compoets of the hypergraph regularity method, i.e., the hypergraph regularity lemma of [33] ad the hypergraph coutig lemma of [22], are techical statemets which we will oly preset later i Sectio 4 (cf. Remar 2.5). The followig socalled removal lemma, however, is a direct cosequece of the regularity method for hypergraphs. Theorem 1.3 (Removal lemma, [12, 22, 32]). For fixed -graph F () o f vertices, suppose H () is a -graph o vertices cotaiig o( f ) (ot ecessarily iduced) copies of F (). The, oe may remove o( ) edges from H () to obtai a subhypergraph G () which is F () -free, i.e., G () cotais o copy of F () at all. Whe F () = K () +1, the removal lemma geeralizes Ruzsa ad Szemerédi s triagle removal lemma (discussed earlier) to -uiform hypergraphs. Fral ad the secod author [10, 26] observed that the the removal lemma (with F () = K () +1 ) implies Szemerédi s theorem (see Theorem 1.4 below with d = 1). Subsequetly,

3 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 3 Solymosi [38, 39] showed that Theorem 1.3 also implies the multidimesioal versio of Szemerédi s theorem, origially due to Fursteberg ad Katzelso [11] (see also [31] for aother cosequece of Theorem 1.3 of similar flavor). Theorem 1.4 (multidimesioal Szemerédi theorem). For fixed itegers l ad d, ay set Z {1,..., } d cotaiig o homothetic copy of {1,..., l} d has size Z = o( d ). I Theorem 1.4, {1,..., } d deotes, as usual, the d-fold cross product of the set {1,..., } with itself. A homothetic copy of {1,..., l} d is ay set of the form a+ c{1,..., l} d, where a {1,..., } d ad c is some positive iteger. 2. Results I this paper, we cosider some extremal hypergraph problems of Turá ad Ramsey flavor. We begi with some problems of Turá-type A Turá-type problem. Geeralizig Turá s problem for hypergraphs, the followig problem was iitiated by Brow, Erdős ad T. Sós [6]. Let f (r) (, v, e) deote the maximum umber of edges i a r-graph o vertices i which o v vertices spa e (or more) edges. Note that the determiatio of f (r) (, v, ( v r) ) = ) is precisely Turá s problem, o which we shall expad i Sectio 2.2. It was first proved by Brow, Erdős ad T. Sós [6] that f (r) (, e(r )+, e) = Θ( ). The same authors ased what happes if, istead of o v = e(r ) + vertices, oe forbids e edges to appear o v +1 = e(r )+ +1 vertices. I particular, they cojectured f (r) (, e(r ) + + 1, e) ca be bouded by o( ). This cojecture was proved for e = r = 3 ad = 2 by Ruzsa ad Szemerédi [36] ad geeralized to arbitrary r with = 2 ad e = 3 by Erdős, Fral ad Rödl [7] ad r > ad e = 3 by Alo ad Shapira [2]. Theorem 1.3 easily implies the upper boud for r > 2 ad e = + 1. We preset the details i Sectio 3. ex(, K (r) v Theorem 2.1. For r > 2, f (r) (, ( + 1)(r + 1), + 1) = o( ). Theorem 2.1 was proved for = 2 by Erdős, Fral ad Rödl [7], ad for = 3 by Sárözy ad Selow [37]. Remar 2.2. I this paper, the iteger otatio is usually reserved for the uiformity of hypergraphs H (), while our otatio f (r) (, v, e) appears to brea with that traditio (sice here, r deotes uiformity). I Theorem 2.1, however, the essetial part of provig the assertio f (r) (, ( + 1)(r + 1), + 1) = o( ), i fact, ivolves appealig to specific auxiliary -uiform hypergraphs H (), where the iitial uiformity r plays less of a rôle. I this sese, we reserve cosistet use of uiformity otatio for later, i the proof, where we feel it is most importat Forbidde families. For a iteger, let F () = {F () i } i I be a give (possibly ifiite) family of -graphs. Let Forb(, F () ) deote the family of all - graphs H () F () i o vertex set {1,..., } cotaiig o sub-hypergraph isomorphic to for all i I. As i the classical Turá problem, set { } ex(, F () ) = max H () : H () Forb(, F () ). Whe F () = {K () l ex(, {K () l } cosists of the sigle clique K () l, determiig ex(, K () l ) = }) is the well-ow Turá problem, where eve the asymptotic for

4 4 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT the case l = 4 ad = 3 remais ope today. For = 2, Turá s formula for these umbers is a cetral result i extremal graph theory. Note that the parameter ex(, K () l ) correspods to f () (, l, ( l ) ) from Sectio 2.1. I the cotext of Turá s problem, however, the ex otatio appears more commoly tha the f otatio, ad so we shall ot brea from this traditio here. Our result i Theorem 2.3 below aims to relate ex(, F () ) with the cardiality Forb(, F () ). Observe that sice all sub-hypergraphs of a fixed H () Forb(, F () ) also belog to Forb(, F () ), we have Forb(, F () ) 2 ex(,f()). We show that this boud is, i a sese, best possible. Theorem 2.3. For every (possibly ifiite) family of -graphs F () = {F () i } i I, we have log 2 Forb(, F () ) = ex(, F () ) + o( ). Theorem 2.3 was proved for = 2 ad F (2) = {K (2) l } by Erdős, Kleitma ad Rothschild [9] ad for geeral F (2) by Erdős, Fral ad Rödl [7]. Theorem 2.3 was proved for = 3 by the first two authors [21]. Bollobás ad Thomaso [4] showed that lim log 2 Forb(, F () ) / ( ) exists for ay family F () ad so Theorem 2.3 provides a combiatorial evaluatio of this limit. We metio that for = 2, a iduced versio of Theorem 2.3 was established by Prömel ad Steger [25] ad by Bollobás ad Thomaso [5]. These results were exteded to = 3 by Kohayaawa ad the first two authors i [18]. Usig the hypergraph regularity method, oe may prove a iduced versio of Theorem 2.3 for geeral 2, ad we hope to address this problem i a forthcomig paper A iduced Ramsey theorem. For a fixed -graph F (), a -graph G () is said to be a iduced Ramsey -graph for F () if every 2-colorig of G () admits a moochromatic sub-hypergraph isomorphic to F () which appears as a iduced sub-hypergraph of G (). Nešetřil ad Rödl [23, 24] ad idepedetly Abramso ad Harrigto [1] proved that every -graph F () has a Ramsey -graph G () for F (). I this paper, we preset aother proof of the iduced Ramsey theorem (based o the hypergraph regularity method). Theorem 2.4. For every iteger 2 ad every -graph F (), there exists a iduced Ramsey -graph G () for F () Orgaizatio of paper. I Sectio 3, we prove Theorem 2.1 usig the removal lemma, Theorem 1.3. While Theorem 2.1 is a cosequece of the removal lemma, we prove Theorem 2.3 ad Theorem 2.4 usig the hypergraph regularity method. I Sectio 4, we preset the hypergraph regularity lemma ad hypergraph coutig lemma. I Sectio 5, we prove Theorem 2.3. I Sectio 6, we prove Theorem 2.4. We coclude the itroductio with the followig remar. Remar 2.5. The compoets of the hypergraph regularity method, the hypergraph regularity lemma ad hypergraph coutig lemma, tae differet forms i the versios [12, 13] ad [22, 33] ad subsequet versios [29] ad [41]. While ay of these versios would suffice to prove the applicatios i this paper, we fid the recet versio of this method due to the secod two authors [29] (based o ideas from [22, 33]) most coveiet for our purposes. We preset these tools i Sectio 4.

5 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 5 3. Proof of Theorem 2.1 We use Theorem 1.3, the removal lemma, to prove Theorem 2.1. I particular, we use the followig corollary of the removal lemma to prove Theorem 2.1. Corollary 3.1. For fixed iteger 2, let -graph H () property that each -tuple K H () The, H () = o( ). o vertices have the belogs to precisely oe copy of the clique K () +1. Proof. Corollary 3.1 follows easily from Theorem 1.3 i the case whe F () cosists of the sigle -clique K () +1 o + 1 vertices. Let H () be give as i the hypothesis of Corollary 3.1. Sice each -tuple K H () belogs to precisely oe copy of K () +1, we see that the umber of such cliques, K +1 (H () ), satisfies K+1 (H () ) 1 = + 1 H() = o( +1 ). (1) Puttig F () = K () +1, Theorem 1.3 the asserts that oe may delete o( ) may - tuples K H () to obtai a K () () +1-free sub-hypergraph H H (). However, sice deletig a -tuple K H () destroys exactly oe clique K () +1, we must have K +1 (H () ) = o( ) ad Corollary 3.1 follows from (1). Proof of Theorem 2.1. Our proof follows the lies of [7, 36, 37], where the earlier established removal lemmas for graphs ad 3-graphs were used to prove the special cases = 2, 3. Let r > 2 be give as i Theorem 2.1. Suppose, o the cotrary, that there exists c = c(r, ) > 0 ad positive iteger 0 = 0 (r,, c) for which f (r) (, ( + 1)(r + 1), + 1) > c (2) holds for all > 0. Let G (r) be a r-graph o > 0 (r,, c) vertices with c may r-tuples with the property that o ( + 1)(r + 1) vertices spa ( + 1) may r-tuples. We shall demostrate that the existece of such G (r) cotradicts Corollary 3.1. We begi by reducig the r-graph G (r) to a r-partite sub-hypergraph G (r). A simple averagig argumet (see, e.g., [8]) implies that the vertex set V (G (r) ) admits a r-partitio V (G (r) ) = V 1 V r for which [ ] G (r) r! V 1,..., V r r r G(r) > r! r r c, (3) where G (r) [V 1,..., V r ] is the sub-hypergraph of G (r) cosistig of all r-tuples R G (r) with R V i = 1 for all 1 i r. For simplicity, set G (r) = G (r) [V 1,..., V r ]. We ow reduce the r-graph G (r) to ( + 1)-graph G (+1) with vertex set V 1 V +1 as follows: for a ( + 1)-tuple K + satisfyig K + V i = 1, 1 i + 1, put K + G (+1) if, ad oly if, K + R for some R G (r). We mae the followig claim. Claim 3.2. G (+1) e G (r) (3) > cr! r r. Proof. The secod iequality immediately follows from (3). To establish the first, we observe that for each K + G (+1), there are at most may r-tuples R G (r) for which K + R (from which Claim 3.2 the follows). Otherwise, if for

6 6 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT some K + G (+1), there exist ( + 1) distict r-tuples R 1,..., R +1 G (r) each cotaiig K +, we would have ( + 1) may r-tuples spaed o +1 i=1 R i (r 1)( + 1) = (r )( + 1) < (r + 1)( + 1) vertices, cotradictig our choice of G (r). We proceed with the followig claim. Claim 3.3. Let K 0 K + 0 G (+1) with K 0 =. There are at most 1 distict ( +1)-tuples K + 1,..., K+ 1 G (+1) for which K + 0 K+ i = K 0, 1 i. Proof. Suppose, o the cotrary, that some fixed K 0 K 0 + G (+1), K 0 =, admits distict ( + 1)-tuples K 1 +,..., K+ G (+1) for which K 0 + K+ i = K 0, 1 i. The, for some R 0, R 1,..., R G (r), we would have ( + 1)-may distict r-tuples R 0 K 0 +, R 1 K 1 +,..., R K + spaed o +1 i=0 R i ( + 1)(r 1) = ( + 1)(r + 1) 1 < ( + 1)(r + 1) vertices, cotradictig our choice of G (r). Claim 3.3 immediately implies that for each K 0 + G (+1), at most (+1)( 1) = 2 1 distict ( + 1)-tuples K 1 +,..., K+ 2 1 G (+1) satisfy K 0 + K+ i =, G (+1) 0 1 i 2 1. As such, the ( + 1)-graph G (+1) cotais a sub-hypergraph of size (+1) G 0 G (+1) Claim 3.2 cr! 2 1 ( 2 1)r r (4) cosistig of ( + 1)-tuples K 0 + G (+1), o two of which overlap i vertices. (+1) Ideed, iteratively costruct ( +1)-graph G 0 by startig with a arbitrary ( + 1)-tuple K 0 + G (+1), deletig all ( + 1)-tuples K + which overlap with K 0 + i (+1) vertices, ad repeatig this procedure util G 0 is produced. We are ow able to coclude the proof of Theorem 2.1. Defie ( + 1)-partite -graph H () o vertex set V 1 V +1 as follows: for a -tuple K 0 satisfyig K 0 V i 1, 1 i + 1, put K 0 H () if, ad oly if, K 0 K 0 + for some ( + 1)-tuple K 0 + (+1) G 0. We mae the followig observatios. (O1) each copy of the clique K () +1 i H() (+1) correspods to a edge of G 0, ad vice-versa; (O2) by costructio of H (), each edge K H () belogs to at least oe copy of the clique K () +1 i H() ; (+1) (O3) by costructio of G 0, each edge K H () belogs to at most oe copy of the clique K () +1 i H() ; (O4) ( ) + 1 H () = G(+1) 0 (4) cr!( + 1) ( 2 1)r r = Ω( ).

7 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 7 Combiig observatios (O2), (O3) ad (O4), we see that H () is a dese -graph whose every edge K H () belogs to precisely oe copy of the clique K () +1. This cotradicts Corollary 3.1 ad hece cocludes the proof of Theorem Regularity method for hypergraphs I this sectio, we preset the hypergraph regularity lemma the hypergraph coutig lemma from [29]. We first preset all eeded defiitios ad otatio i Sectio 4.1. I Sectio 4.2, we state both lemmas Defiitios. We start with some basic cocepts ad otatio Basic cocepts. For itegers l j 1, the otatio [l] deotes the set of itegers {1,..., l} ad [l] j = ( ) [l] j deotes the set of all uordered j-tuples from [l]. I this paper l-partite, j-uiform hypergraphs play a special rôle, where j l. Give vertex sets V 1,..., V l, we deote by K (j) l (V 1,..., V l ) the complete l-partite, j- uiform hypergraph (i.e., the family of all j-elemet subsets J i [l] V i satisfyig V i J 1 for every i [l]). If V i = m for every i [l], the a (m, l, j)- cylider H (j) o V 1 V l is ay subset of K (j) l (V 1,..., V l ). The vertex partitio V 1 V l is a (m, l, 1)-cylider H (1). (This defiitio may seem artificial right ow, but it will simplify later otatio.) For j i l ad set Λ i [l] i, we deote by H (j) [Λ i ] = H (j)[ ] λ Λ i V λ the sub-hypergraph of the (m, l, j)-cylider H (j) iduced o λ Λ i V λ. For a (m, l, j)-cylider H (j) ad a iteger 2 j i l, we deote by K i (H (j) ) the family of all i-elemet subsets of V (H (j) ) which spa complete sub-hypergraphs i H (j) of order i. For 1 i l, we deote by K i (H (1) ) the family of all i-elemet subsets of V (H (1) ) which cross the partitio V 1 V l, i.e., I K i (H (1) ) if, ad oly if, I V s 1 for all 1 s l. For 2 j i l, K i (H (j) ) is the umber of all copies of K (j) i i H (j). Give a (m, l, j 1)-cylider H (j 1) ad a (m, l, j)-cylider H (j), we say H (j 1) uderlies H (j) if H (j) K j (H (j 1) ). This brigs us to oe of the mai cocepts of this paper, the otio of a complex. Defiitio 4.1 ((m, l, h)-complex). Let m 1 ad l h 1 be itegers. A (m, l, h)-complex H is a collectio of (m, l, j)-cyliders {H (j) } h j=1 such that (a ) H (1) is a (m, l, 1)-cylider, i.e., H (1) = V 1 V l with V i = m for i [l], ad (b ) H (j 1) uderlies H (j) for 2 j h, i.e., H (j) K j (H (j 1) ). We sometimes shorte the termiology (m, l, h)-complex to (l, h)-complex, whe the cardiality m = V 1 = = V s is t of primary cocer Relative desity ad hypergraph regularity. We begi by defiig a relative desity of a j-uiform hypergraph w.r.t. (j 1)-uiform hypergraph o the same vertex set. Defiitio 4.2 (relative desity). Let H (j) be a j-uiform hypergraph ad let H (j 1) be a (j 1)-uiform hypergraph o the same vertex set. We defie the desity of H (j) w.r.t. H (j 1) as d ( H (j) H (j 1) ) H (j) K j(h (j 1) ) if Kj (H = K j(h (j 1) ) (j 1) ) > 0 0 otherwise.

8 8 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT We also defie a otio of regularity for (m, j, j)-cyliders w.r.t. some uderlyig (m, j, j 1)-cyliders. Defiitio 4.3 ((ε, d)-regular). Let reals ε > 0 ad d 0 be give alog with a (m, j, j)-cylider H (j) ad uderlyig (m, j, j 1)-cylider H (j 1). We say H (j) is (ε, d)-regular w.r.t. H (j 1) if wheever Q (j 1) H (j 1) satisfies Kj (Q (j 1) ) ε Kj (H (j 1) ), the d ( H (j) Q (j 1) ) = d ± ε. Before cotiuig, we pause for the followig remar. Remar 4.4. We compare the otio of regularity i Defiitio 4.3 for j = 2 with the traditioal defiitio of a ε-regular pair (give i the begiig of the Itroductio). The (m, 2, 2)-cylider H (2) is, i the traditioal termiology, a bipartite graph. The uderlyig (m, 2, 1)-cylider H (1) is the bipartitio of H (2), writte here as H (1) = V 1 V 2 where V 1 = V 2 = m. The sub-cylider Q (1) V 1 V 2 is a subset of vertices, which we could write as Q (1) = V 1 V 2, where V 1 V 1 ad V 2 V 2. The assumptio of Defiitio 4.3 sayig K 2 (Q (1) ) ε K 2 (H (1) ) is idetical to sayig V 1 V 2 ε V 1 V 2. As such, the defiitio esures d(h (2) Q (1) ) = d ± ε, or equivaletly, d(h (2) Q (1) ) d < ε. The quatity d(h (2) Q (1) ) is the same as d H (2)(V 1, V 2). The costat d is ot ecessarily the desity d(h (2) H (1) ), but it is, of course, close to it. There is oly oe real differece, therefore, betwee the otio of graph regularity give i Defiitio 4.3 whe j = 2 ad the traditioal defiitio of a ε-regular pair. I the traditioal defiitio, we would assume that the subsets V 1 V 1, V 2 V 2 idividually satisfy the coditios V 1 ε V 1 ad V 2 ε V 2. I Defiitio 4.3, we assume the product V 1 V 2 satisfies the sigle coditio V 1 V 2 ε V 1 V 2. Quite obviously, however, these two otios are equivalet: if H (2) is (ε, d)-regular w.r.t. H (1), the H (1) is a ε-regular pair, ad if H (1) is a ε-regular pair, the H (2) is (ε 2, d(h (2), H (1) ))-regular w.r.t. H (1). We ow exted the otio of (ε, d)-regularity to (m, l, j)-cyliders H (j). Defiitio 4.5 ((ε, d)-regular cylider). We say a (m, l, j)-cylider H (j) is (ε, d)-regular w.r.t. a (m, l, j 1)-cylider H (j 1) if for every Λ j [l] j, the restrictio H (j) [Λ j ] = H (j)[ λ Λ j V λ ] is (ε, d)-regular w.r.t. the restrictio H (j 1) [Λ j ] = H (j 1)[ λ Λ j V λ ]. We ow exted the otio of (ε, d)-regularity from cyliders to complexes. Defiitio 4.6 ((ε, d)-regular complex). Let ε be a positive real ad let d = (d 2,..., d h ) be a vector of o-egative reals. We say a (m, l, h)-complex H = {H (j) } h j=1 is (ε, d)-regular if H(j) is (ε, d j )-regular w.r.t. H (j 1) for every j = 2,..., h Partitios. The regularity lemma for -uiform hypergraphs provides a wellstructured family of partitios P = {P (1),..., P ( 1) } of vertices, pairs,..., ad ( 1)-tuples of some vertex set. We ow discuss the structure of these partitios recursively, followig the approach of [33]. Let be a fixed iteger ad V be a set of vertices. Let P (1) = {V 1,..., V P (1) } be a partitio of V. For every 1 j P (1), let Cross j (P (1) ) be the family of all crossig j-tuples J, i.e., the set of j-tuples which satisfy J V i 1 for every V i P (1).

9 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 9 Suppose that partitios P (i) of Cross i (P (1) ) for 1 i j 1 have bee defied. The for every (j 1)-tuple I i Cross j 1 (P (1) ), there exist a uique P (j 1) = P (j 1) (I) P (j 1) so that I P (j 1). For every j-tuple J i Cross j (P (1) ), we defie the polyad of J ˆP (j 1) (J) = {P (j 1) (I): I [J] j 1}. I other words, ˆP(j 1) (J) is the uique set of j partitio classes of P (j 1) each cotaiig a (j 1)-subset of J. Observe that ˆP (j 1) (J) ca be viewed as a (j, j 1)- cylider, i.e., a j-partite, (j 1)-uiform hypergraph. More geerally, for 1 i < j, we set ˆP (i) (J) = {P (i) (I): I [J] i} ad P(J) = { ˆP(i) (J) } j 1 i=1. (5) Remar 4.7. I this paper, we use P (j), read script P, to deote the partitio of j-tuples. Partitio classes P (j) P (j) (which are j-uiform hypergraphs o []) are deoted with calligraphic P. Uios of special sub-collectios of j-graphs P (j) (which we call polyads) are deoted with calligraphic P equipped with a hat. Next, we defie ˆ P (j 1), the family of all polyads ˆ P (j 1) = { ˆP(j 1) (J): J Cross j (P (1) ) }. Note that ˆP (j 1) (J) ad ˆP (j 1) (J ) are ot ecessarily distict for differet j- tuples J ad J. We view P ˆ(j 1) as a set ad, cosequetly, {K j ( ˆP (j 1) ): ˆP(j 1) P ˆ (j 1) } is a partitio of Cross j (P (1) ). The structural requiremet o the partitio P (j) of Cross j (P (1) ) is P (j) {K j ( ˆP (j 1) ): ˆP(j 1) ˆ P (j 1) }, (6) where deotes the refiemet relatio of set partitios. I other words, we require that the set of cliques spaed by a polyad i P ˆ(j 1) is sub-partitioed i P (j) ad every partitio class i P (j) belogs to precisely oe polyad i P ˆ(j 1). Note that (6) implies (iductively) that P(J) defied i (5) is a (j, j 1)-complex. O a related ote, we shall ofte drop the argumet J Cross j (P (1) ) from the otatio ˆP (j 1) (J) (as the families P with which we wor always satisfy K j ( ˆP (j 1) ) ). Throughout this paper, we wat to cotrol the umber of partitio classes i P (j), ad more specifically, over the umber of classes cotaied i K j ( ˆP (j 1) ) for a fixed polyad ˆP (j 1) P ˆ(j 1). We mae this precise i the followig defiitio. Defiitio 4.8 (family of partitios). Suppose V is a set of vertices, 2 is a iteger ad a = (a 1,..., a 1 ) is a vector of positive itegers. We say P = P( 1, a) = {P (1),..., P ( 1) } is a family of partitios o V, if it satisfies the followig: (i ) P (1) is a partitio of V ito a 1 classes, (ii ) P (j) is a partitio of Cross j (P (1) ) satisfyig: ad P (j) refies {K j ( ˆP (j 1) ): ˆP(j 1) P ˆ(j 1) } { P (j) P (j) : P (j) K j ( ˆP (j 1) ) } = aj for every ˆP(j 1) P ˆ(j 1). Moreover, we say P = P( 1, a) is t-bouded, if max{a 1,..., a 1 } t.

10 10 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT It is easy to see that for a t-bouded family of partitios P ad a iteger 2 j 1, we have ( ) j 1 P ˆ(j 1) a1 = a h) (j h t 2t. (7) j We cotiue with a few fial defiitios eeded to state the hypergraph regularity lemma ad correspodig coutig lemma Regular partitios. The followig defiitio describes some of the structure the regularity lemma shall provide. Defiitio 4.9 ((η, ε, a)-equitable). Suppose V is a set of vertices, η ad ε are positive reals, a = (a 1,..., a 1 ) is a vector of positive itegers where a 1 divides. We say a family of partitios P = P( 1, a) o V (as defied i Defiitio 4.8) is (η, ε, a)-equitable if it satisfies the followig: h=2 (a ) [V ] \ Cross (P (1) ) η ( ), (b ) P (1) = {V i : i [a 1 ]} is a equitable vertex partitio, i.e., V i = V /a 1 for i [a 1 ], ad (c ) for every K Cross (P (1) ) the (/a 1,, 1)-complex P(K) (see (5)) is (ε, (1/a 2,..., 1/a 1 ))-regular. To describe the remaiig structure of the regularity lemma, we exted Defiitio 4.5. Defiitio 4.10 ((δ, d, r)-regular). Let δ ad d be positive reals ad r be a positive iteger. Suppose H ( 1) is a ( 1)-graph ad H () is a -graph, both of which share the same vertex set. We say H () is (δ, d, r)-regular w.r.t. H ( 1) if for every collectio Q ( 1) = {Q ( 1) 1,..., Q ( 1) r } of ot ecessarily disjoit sub-hypergraphs of H ( 1) satisfyig K (Q ( 1) i ) > δ K (H ( 1) ), we have i [r] H () i [r] K (Q ( 1) i ) i [r] K (Q ( 1) i ) = d ± δ. We write (δ,, r)-regular to mea ( δ, d ( H () H ( 1) ), r ) -regular. We eed oe last defiitio to state the regularity lemma. Defiitio 4.11 ((δ, r)-regular w.r.t. P). Suppose δ is a positive real ad r is a positive iteger. Let H () be a -uiform hypergraph with vertex set V ad P = P( 1, a) be a family of partitios o V. We say H () is (δ, r)-regular w.r.t. P, if {K ( ˆP ( 1) ): ˆP( 1) P ˆ( 1) ad H () is ot (δ,, r)-regular w.r.t. ˆP ( 1)} ( ) V δ.

11 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD Hypergraph regularity lemma ad coutig lemma. The regularity lemma of [29] is give as follows. Theorem 4.12 (Regularity lemma). Let 2 be a fixed iteger. For all positive costats η ad δ ad fuctios r : N 1 N ad δ : N 1 (0, 1] there are itegers t Thm.4.12 ad Thm.4.12 so that the followig holds. For every -uiform hypergraph H () satisfyig V (H () ) = Thm.4.12 ad t Thm.4.12! dividig, there exists a family of partitios P = P( 1, a P ) so that (i ) P is (η, δ(a P ), a P )-equitable ad t Thm bouded; (ii ) H () is (δ, r(a P ))-regular w.r.t. P. The followig hypergraph coutig lemma correspods to Theorem Theorem 4.13 (Coutig lemma). For all itegers l 2 ad positive costats γ > 0 ad d > 0, there exists δ > 0 such that for all itegers a 1,..., a 2, there are a costat δ > 0 ad positive itegers r ad m 0 so that the followig holds. Suppose (i ) R = {R (j) } 1 j=1 is a (δ, (1/a 2,..., 1/a 1 ))-regular (m, l, 1)-complex with m m 0, ad (ii ) for every Λ [l], the -graph H () K (R ( 1) ) is (δ, d Λ, r)-regular w.r.t. R ( 1) [Λ ] for some d Λ d. The K l (H () ) (1 γ)d ) (l 1 j=2 ( 1 a j 5. Proof of Theorem 2.3 ) ( l j) m l. The mai idea i provig Theorem 2.3 is ot difficult, but sice it ivolves appealig to the regularity lemma ad coutig lemma for hypergraphs, its appearace is techical. We therefore begi this sectio by setchig this mai idea i the (more trasparet) case of graphs, followig the wor of [7]. I the followig outlie, we restrict our attetio to the special case whe F (2) = {K (2) 3 } cosists of the (sigle) triagle K 3 = K (2) 3. We metio that, if we focus our attetio to whe F (2) cosists of a sigle graph, our choice here of K 3 maes little differece i the argumet. However, restrictig our attetio to whe F (2) cosists of oly fiitely may graphs frees us from oe detail which is similarly techical for graphs as it is for hypergraphs The graph case with F (2) = K 3. Fix ν > 0. We setch the proof that Forb(, K3 ) 2 ex(,k 3)+ν 3 holds for all large itegers. The mai compoets of the proof are the Szemerédi regularity lemma, Theorem 1.1, ad the coutig lemma (for graphs), Fact 1.2. We begi by regularizig every graph G = G (2) i the collectio Forb(, K 3 ). To that ed, we pic some small 0 < ε = ε(ν) ν (we wo t determie a formula for ε at this time sice we pla to bypass, i this outlie, the calculatios usig this formula) ad large iteger t 0 = t 0 (ν) 1/ν. As we mae these choices, we also pic a auxiliary costat ε d 0 ν which is small w.r.t. ν but large w.r.t. ε. Theorem 1.1 guaratees a iteger T 0 = T 0 (ε, t 0 ) so that, with large,

12 12 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT every graph G Forb(, K 3 ) admits a ε-regular, t G -equitable partitio V (G) = V1 G Vt G G where t 0 t G T 0. For G Forb(, K 3 ), we shall write P G for the ε-regular, t G -equitable partitio V (G) = V1 G Vt G G, t 0 t G T 0, obtaied above. We fix, for each G Forb(, K 3 ), the partitio P G ow obtaied (ad if G admits multiple such, we simply pic oe, arbitrarily). I all that follows, = (ν, d 0, ε, t 0, T 0 ) is sufficietly large w.r.t. all the costats metioed above. We first decompose Forb(, K 3 ) ito equivalece classes. We say two graphs G 1 ad G 2 Forb(, K 3 ) are equivalet if, ad oly if, P G1 = P G2. (I other words, the ε-regular partitios P G1 ad P G2 fixed above split the vertices {1,..., } i precisely the same way.) Let Forb(, K 3 ) = Π 1 Π N be the partitio of Forb(, K 3 ) associated with this equivalece relatio. The Forb(, K 3 ) = N a=1 Π a, ad clearly, there are at most N T0 = 2 o(2) partitios of the vertices {1,..., }. Thus, it suffices to estimate Π a for a arbitrary idex 1 a N. For the remaider of this outlie, fix 1 a N. There is a commo partitio P a of {1,..., } that every graph G Π a admits as its fixed ε-regular partitio P G. We write P a as V 1 V t, where t 0 t T 0. Now, for a fixed G Π a, we shall record for which pairs (V i, V j ) of the partitio P a the graph G is dese ad regular. More formally, for 1 i < j t, write x G = (x G ij : 1 i < j t), where { x G 1 if d G (V i, V j ) d 0 ad V i, V j is ε-regular w.r.t. G, ij = 0 otherwise. For fixed x {0, 1} (t 2), we set Πa (x) = {G Π a : x G = x} ad observe Sice there are oly 2 (t 2) 2 T 2 0 Π a = { Π a (x) : x {0, 1} (t 2) }. = 2 O(1) = 2 o(2) vectors x {0, 1} (t 2), it suffices to estimate Π a (x) for a fixed but arbitrary x {0, 1} (t 2). With x fixed, ad a fixed before, we ow defie D a (x) as the graph with vertex set {1,..., t} ad edges {i, j}, 1 i < j t, correspodig to whe the pair (V i, V j ) is dese ad regular w.r.t. every graph G Π a (x), i.e., whe x ij = 1. If we ca show the it will be easy to show D a (x) ex(t, K 3 ) (8) Π a (x) 2 ex(,k3)+ ν 2 2. (9) Establishig the implicatio (8) = (9) is stadard, ad so we oly highlight it here. Ideed, usig stadard cosideratios of ε-regular partitios, oe may easily show that for ay G Π a (x) { {v i, v j } E(G): v i V i, v j V j, either i = j or x ij = 0 } ( ) 1 < + ε + d 0 2 ν2 (10) t 2 0 where the last iequality holds by virtue of the fact that we chose 1/t 0, ε, ad d 0 much smaller tha ν. Hece there are essetially 2 ν 2 2 choices for the subgraphs of graphs G Π a (x) iduced o vertex classes V i (i = 1,..., t) ad o pairs (V i, V j )

13 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 13 with x ij = 0. The umber of subgraphs o pairs (V i, V j ) with x ij = 1 is (igorig precise error calculatios) approximately 2 P {i,j} Da(x) Vi Vj 2 P {i,j} Da(x) 2 t 2 (8) 2 ex(,k3) (11) where the last asymptotic employs (8) ad maes use of the fact that ex(t, K 3 )/ ( t 2) ex(, K 3 )/ ( 2) wheever t ad are large (recall t t0, where we piced t 0 large ). Sice every graph G Π a (x) behaves idetically o the commo partitio P a, every graph G Π a (x) must cosist of oe of the (essetially) 2 ν 2 2 may subgraphs couted i (10), ad oe of the (essetially) 2 ex(,k3) subgraphs couted i (11). This completes the setch of (8) = (9). We fiish the preset outlie by provig (8), ad to that ed, we use the coutig lemma, Fact 1.2. Ideed, if D a (x) > ex(t, K 3 ), the D a (x) cotais a copy of the triagle K 3. Let i, j, deote the vertices of this triagle (which correspod to the vertex classes V i, V j, V of the partitio P a ) ad fix ay graph G 0 Π a (x). By defiitio of D a (x), each of the pairs {V i, V j }, {V j, V } ad {V i, V } are ε-regular w.r.t. G 0 ad also satisfy d G0 (V i, V j ), d G0 (V j, V ), d G0 (V i, V ) d 0. By the coutig lemma, Fact 1.2, the graph G 0 cotais at least d 3 0(/t) 3 > 0 may triagles K 3, which cotradicts that G 0 Forb(, K 3 ). This completes the outlie. Before proceedig to the actual proof of Theorem 2.3, we mae the followig remar. Remar 5.1. As we metioed before, oe has to wor a little harder, whether for graphs or hypergraphs, whe the set F () cosists of ifiitely may elemets rather tha fiitely may. These details were ot addressed i our outlie, but are addressed i our proof of Theorem 2.3. As well, i our proof of Theorem 2.3, we shall defie a -graph D α (x) i (26) which is a aalogue to the graph D a (x) (cf. (8)). For reasos we do ot metio here, we defie D α (x) i a slightly differet way tha we defied D a (x). I the ed, however, the ivocatio of the coutig lemma will be precisely the same as i the outlie above Settig up the proof of Theorem 2.3. I our proof of Theorem 2.3, we use the followig otatio. For a iteger ad a family of -graphs F (), set ẽx(, F () ) = ex(, F() ) ( ). It is well ow (see [16]) that the sequece (ẽx(, F () )) =1 is o-icreasig, ad hece, π(f () ) = lim ẽx(, F() ) (12) exists. Note that whe π(f () ) = 0 the assertio of Theorem 2.3 is trivial. Ideed, Forb(, F () ) o( ) Heceforth, we shall assume π(f () ) > 0. s=0 (( ) ) = 2 o(). s

14 14 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT It suffices to prove Theorem 2.3 for divisible by a fixed but arbitrary iteger T. I particular, suppose that, for fixed ν > 0 ad fixed iteger T, for every iteger m > m 0 (, ν, T ), we have Forb(mT, F () ) 2 ex(mt,f() )+ν(mt ). The it easily follows that for all itegers > 0 (, ν, T ), Forb(, F () ) 2 ex(,f() )+2ν. Ideed, for a iteger, write (m 1)T < mt for some iteger m. The, with m ad sufficietly large, we have log 2 Forb(, F () ) log 2 Forb(mT, F () ) ex(mt, F () ) + ν(mt ) ( ) ( ) mt mt = ẽx(mt, F () ) + ν(mt ) π(f () ) + ν(mt ) + o((mt ) ) ( ) ( ) + T π(f () ) + ν( + T ) + o(( + T ) ) = π(f () ) + ν + o( ) ( ) ( ) ẽx(, F () ) + ν + o( ) ẽx(, F () ) + 2ν = ex(, F () ) + 2ν, where the ext to last iequality follows from the sequece (ẽx(s, F () )) s=1) beig o-icreasig with limit π(f () ). We ow prove that for every ν > 0, there exist itegers T = T (ν) ad 0 = 0 (ν, T ) so that for every 0 divisible by T, ( ) log 2 Forb(, F () ) ex(, F () ) + ν. (13) As our proof depeds o Theorems 4.12 ad 4.13, we first discuss a sequece of auxiliary costats Costats. Let ν > 0 be give. Let f 0 N be sufficietly large so that ẽx(f 0, F () ) < π(f () ) + ν 8. (14) Choose 0 < η = d 0 < 1/9 so that (1 η) 1/( 1) 1 1 f 0 ad 4d 0 log 2 e 3d 0 ν 4 (15) (ote that the last iequality uses x log 2 x 0 as x 0 + ). For fixed itegers f 0 ad ad costats γ = 1/2 ad d = d 0, let δ = δ (4.13) (f 0,, 1/2, d 0 ) (16) be the costat guarateed by Theorem We may assume, without loss of geerality, that δ d 0. (17) For positive iteger variables y 1,..., y 2, let δ(y 1,..., y 2 ) = δ (4.13) (f 0,, 1/2, d 0, δ, y 1,..., y 2 ) (18) r(y 1,..., y 2 ) = r (4.13) (f 0,, 1/2, d 0, δ, y 1,..., y 2 ) (19) be the fuctios guarateed by Theorem 4.13.

15 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 15 We ow defie further costats i terms of the regularity lemma, Theorem With iput parameters η ad δ ad fuctios 1 δ(y 1,..., y 2 ) ad r(y 1,..., y 2 ) defied above, Theorem 4.12 guaratees iteger costats t = t (4.12) (η, δ, δ, r) ad 0 = (4.12) (η, δ, δ, r). (20) The costat T metioed i (13) is set to be T = t!. Now, for > 0 divisible by T ad sufficietly large, we verify (13) Proof of (13). Accordig to Theorem 4.12, every -graph G () o vertices ( defied above) admits a (η, δ(a P ), a P )-equitable t-bouded family of partitios P with respect to which G () is (δ, r(a P ))-regular. As such, for each G () Forb(, F () ), we may associate a family of partitios P G () (if G () admits multiple such partitios, we simply choose oe of them). Accordigly, we may impose a equivalece relatio o Forb(, F () ) accordig to the followig rule: for G (), G () Forb(, F () ), G () G () P G () = P eg (). (21) Let Forb(, F () ) = Π 1 Π N be the partitio of Forb(, F () ) iduced by. To prove (13), we first see to boud the parameter N = N(). Clearly, N is at most the umber of t-bouded families of partitios o the vertex set []. For a fixed vector a = (a 1,..., a 1 ), there are at most 1 j) j=1 a( j families of partitios P( 1, a) o the vertex set []. Cosequetly, { 1 } a j) ( j : 1 a j t for j = 1,..., 1 t 1 t P 1 j=1 ( j) = 2 O( 1). N a j=1 (22) We ow see to boud Π α for every α = 1,..., N. Fix 1 α N ad, correspodigly, family of partitios P α = {P α (1),..., P α ( 1) }, i.e., the family associated to every G () Π α. With each G () Π α, we associate the vector ( x G () = x : ˆP( 1) ˆ ) P ( 1) ( 1) ˆP {0, 1} α, (23) ˆP( 1) α where, for fixed ˆP ( 1) P ˆ α ( 1), 1 if d(g () ˆP ( 1) ) d 0 ad x = G ˆP( 1) () is (δ,, r(a Pα ))-regular w.r.t. ˆP( 1), 0 otherwise. From (7) ad the t-boudedess of the family P α, (24) {x G () : G () Π α } 2 t2 = O(1). (25) 1 Note that the iput fuctios fuctios δ(y 1,..., y 2 ) ad r(y 1,..., y 2 ) have 2 variables while Theorem 4.12 would allow us to cosider 1 variables. I particular, Theorem 4.12 would allow us to iclude a variable y 1 correspodig to the umber of vertex classes the output family of partitios P will have. We have o eed for this feature i our argumet here, so we hold the variable y 1 costat.

16 16 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT With α [N] fixed, fix vector x {0, 1} We prove the followig lemma. ( 1) ˆP α ad defie Π α (x) = { G () Π α : x G () = x }. Lemma 5.2. log 2 Π α (x) ex(, F () ) + ν 2 ( ). Lemma 5.2, combied with (22) ad (25), easily implies (13) (ad hece, Theorem 2.3). Ideed Forb(, F () ) = N Π α = α=1 N Π α (x) α=1 x 2 O( 1) O(1) 2 ex(,f() )+ ν 2 ( ) 2 ex(,f () )+ν( ) where the last iequality holds for sufficietly large. We ow proceed to prove Lemma Proof of Lemma 5.2. Fix α [N] ad, correspodigly, P α = P α ( 1, a Pα ) with a Pα = (a 1,..., a 1 ) ad fix x = (x : ˆP( 1) P ˆ( 1) ˆP( 1) α ). Defie D α (x) to be the set of -tuples K Cross (P α (1) ) for which each G () Π α (x) is dese ad regular w.r.t. ˆP ( 1) (K): D α (x) = {K ( ˆP } ( 1) ): x = 1 (cf. (24)). (26) ˆP( 1) We mae the followig claim. Claim 5.3. D α (x) (ẽx(, F () ) + ν 4 )( ). Our proof of Claim 5.3 is based o the coutig lemma, Theorem O the other had, Lemma 5.2 is a simple cosequece of Claim 5.3. As such, we go ahead ad assume Claim 5.3, for the momet, ad fiish the proof of Lemma 5.2, before we verify Claim 5.3. Fiishig the proof of Lemma 5.2, ote that every edge K ( ) [] \Dα (x) satisfies that either (I) K is o-crossig i P (1), (II) or x ˆP( 1) (K) = 0, i.e., by (24), polyad ˆP ( 1) (K) is either sparse or irregular (for every G () Π α (x)). However, sice every G () Π α (x) is (δ, r(a P ))-regular w.r.t. (η, δ(a Pα ), a Pα )- equitable family P α, the umber of edges K ( ) [] satisfyig (I) or (II) is at most ( ) (15), (17) ( ) (η + δ + d 0 ) 3d 0. (Ideed, the equitability of family P α esures that there are at most η ( ) ocrossig edges. The fact that every G () Π α (x) is (δ, r(a P ))-regular w.r.t. family P α esures that at most δ ( ) may -tuples belog to irregular polyads. Fially, sparse polyads (with desity smaller tha d 0 ), i total, ca oly give rise to at most d 0 ( ) may -tuples.)

17 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 17 Now, every G () Π α (x) ca be writte as a disjoit uio G () = G () 1 G () 2 where G () 1 D α (x) ad G () 2 3d 0( ). As such, 3d 0( ) (( ) Claim 5.3 Π α (x) 2 Dα(x) ) j j=0 which implies (with large) log 2 Π α (x) ( ẽx(, F () ) + ν 4 + 4d 0 log as promised by Lemma 5.2. It ow oly remais to prove Claim Proof of Claim 5.3. Let α [N] ad x {0, 1} set A Cross a1 (P α (1) ), defie auxiliary -graph ( ) 2 ( eex(,f() )+ ν 4 )( ) e 3d0( ), 3d 0 e )( ) (15) ex(, F () ) + ν 3d 0 2 ( 1) ˆP α Dese(A) = { K ( A ) : x ˆP( 1) (K) = 1 (cf. (24))}. ( ), be fixed. For crossig Double-coutig pairs (A, K) where K Dese(A) ad A Cross a1 (P α (1) ) yields Dα (x) ( ) a1 = Dese(A). (27) a 1 A Cross a1 (P (1) α ) As such, we may ifer Claim 5.3 from the the followig assertio: ( max{ Dese(A) : A Cross a1 (P α (1) )} < ẽx(a 1, F () ) + ν 8 Ideed, sice Cross a1 (P (1) α D α (x) ( < ẽx(a 1, F () ) + ν 8 ) ( a 1 ). (28) ) = (/a 1 ) a1, we combie (27) ad (28) to say ) ( ) ( ) a 1 ( ẽx(a 1, F () ) + ν ) ( ). 8 Sice 2 a 1 f 0 (where f 0 is give i (14)) ad the sequece (ẽx(s, F () )) s=1 is o-icreasig with limit π(f () ) (see (12)), we have Dα (x) ( < ẽx(f 0, F () )+ ν ) ( ) (14) ( < π(f () )+ ν ) ( ) ( ẽx(, F () )+ ν ) ( ) Thus, it remais to prove the assertio i (28). Proof of (28). O the cotrary, suppose there exists A Cross a1 (P α (1) ) so that ( Dese(A) ẽx(a 1, F () ) + ν ) ( ) a 1. (29) 8 As such, we claim there must also exist B ( A f 0 ) (see (14)) such that the subhypergraph Dese B (A) of Dese(A) iduced o B cotais at least ex(f 0, F () ) It is easy to see a1 f 0. Ideed, sice P α is a (η, δ(a Pα ), a Pα )-equitable family of partitios ad sice Cross (P α (1) ) = `a 1 ( ) a, we have 1 1 η Cross (P α (1) 1 ) 1 1 «1 a 1 where the last iequality holds with sufficietly large. The assertio a 1 f 0 the follows from our choice of η i (15), i.e., 1 η (1 f 1 0 ) 1. a 1

18 18 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT edges. Ideed, supposig otherwise, the umber M of pairs (K, B), K ( B ), B ( ) A f 0, would, o the oe had, satisfy ( ) ( )( ) a1 M ex(f 0, F () ) = ẽx(f 0, F () f0 a1 ). (30) O the other had, by the choice of A i (29), ( M ẽx(a 1, F () ) + ν 8 f 0 ) ( a 1 f 0 )( ) a1. f 0 The mootoicity of the sequece (ẽx(s, F () ) : s 1) the gives ( M ẽx(a 1, F () ) + ν ) ( )( ) a 1 a1 8 f 0 ( π(f () ) + ν ) ( )( ) ( )( ) a 1 a1 (14) > ẽx(f 0, F () a1 a1 ), 8 f 0 f 0 cotradictig (30). Fix B ( ) A f 0 for which the f0 -vertex sub-hypergraph Dese B (A) of Dese(A) iduced o B cotais at least ex(f 0, F () )+1 edges. The, there exists F () F () so that its copy F () 0 appears as a sub-hypergraph of Dese B (A). I order to derive a cotradictio from our assumptio i (29), we use the coutig lemma, Theorem 4.13, to fid a copy of the same F () F () i ay (ad every) G () Π α (x). Sice Π α (x) Forb(F () ), we have a immediate cotradictio. Set To that ed, fix G () Π α (x) ad let F = V (F () 0 ) B. For each K ( F ), set { G () K ( ˆP ( 1) (K)) if K F () H () K = K ( ˆP ( 1) (K)) H () = { H () K otherwise. : K ( F ) }. With H () defied above, observe that every elemet of K f (H () ), f = F correspods to a copy of F () appearig as a sub-hypergraph of G (). If we show K f (H () ) > 0, the we derive a cotradictio, ad hece, (28) follows. To show K f (H () ) > 0, we apply the coutig lemma, Theorem 4.13, to H () ad Q = {Q (j) } 1 j=1 where Q(j) = { P (j) (J): J ( F j ) } for j = 1,..., 1. We first chec that the assumptios of Theorem 4.13 are met by H () ad Q: (1) Sice P α is a (η, δ(a P ), a P )-equitable family, the (/a 1, f, 1)-complex Q is (δ(a P ), (1/a 2,..., 1/a 1 ))-regular. Moreover, we chose the fuctio δ i (18) appropriately for a applicatio of Theorem 4.13; (2) For each K F () 0 Dese B (A) Dese(A), the defiitio of x i (23) guaratees that H () K = G() K ( ˆP ( 1) (K)) is (δ,, r(a P ))-regular w.r.t. ˆP ( 1) (K) Q ( 1) ad that d(g () ˆP ( 1) (K)) d 0. We ote that δ ad r were chose i (16) ad (19) appropriately for a applicatio of Theorem 4.13; (3) For each K ( ) F () \ F 0, the -graph H () K = K ( ˆP ( 1) (K)) is easily see to be (ε, 1, s)-regular w.r.t. ˆP( 1) (K) for every ε > 0 ad s N. As such, H K is (δ, 1, r(a P ))-regular w.r.t. ˆP( 1) (K). 0,

19 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 19 Hece, we ca apply the hypergraph coutig lemma to H () ad Q. We coclude K f (H () ) 1 1 ( ) ) 1 ( f j) ( ) f 2 d(f a j a 1 2 d(f0 ) ( ) 1 ( f 0j ) ( ) f0 0 > 0. a j a 1 This proves (28). j=2 6. Proof of Theorem 2.4 j=2 Theorem 2.4 is a simple cosequece of the followig lemma. Lemma 6.1. Let -graph F () o f vertices be give. For every c > 0, there exist ε > 0 ad itegers r, T ad 0 so that a give -graph G () o vertex set [] = {1,..., }, with 0 ad divisible by T, is a iduced Ramsey -graph for F () wheever the followig coditios are met: (i ) K s (G () ) c ( ) s where s = R () (f, f) is the Ramsey umber for K () f ; (ii ) G () is (ε, d(g () P ( 1) ), r)-regular, d(g () P ( 1) ) [ 1 4, 3 4 ], w.r.t. every ( 1)-graph P ( 1) ( [] 1) which satisfies K (P ( 1) ) / log. Lemma 6.1 implies Theorem 2.4. Ideed, it is easy to verify that, with probability tedig to 1 as, i.e., asymptotically almost surely (a.a.s.), the biomial radom -graph G () (, 1/2) satisfies the hypothesis of Lemma 6.1 with c = (1/2) (s ) 1 ad with arbitrary choices of ε > 0 ad itegers r ad T. I particular, Chebyshev s iequality verifies that G () (, 1/2) satisfies (i ), a.a.s. For completeess, we verify i the Appedix (see Fact A.1) that G () (, 1/2) satisfies a.a.s. (ii ). The goal of this sectio is, therefore, to prove Lemma 6.1. As our proof depeds o Theorems 4.12 ad 4.13, we agai first discuss a sequece of auxiliary costats Costats. Let -graph F () o f vertices be give. Set, as i the hypothesis of Lemma 6.1, s = R () (f, f). (31) As i the hypothesis of Lemma 6.1, let c > 0 be give. We defie ε > 0 ad itegers r ad t i terms of Theorem 4.12 ad As i Theorem 4.13, put l = f, γ = 1/2 ad d = 1/8 ad let δ (4.13) = δ (4.13) (f,, 1/2, d ) be the costat guarateed by Theorem Set η = δ = mi { 1 2 δ(4.13), c 4 For positive iteger variables y 1,..., y 2, let ( s ) 1 } (32) δ(y 1,..., y 2 ) = δ (4.13) (f,, 1/2, d, y 1,..., y 2 ), (33) r(y 1,..., y 2 ) = r (4.13) (f,, 1/2, d, y 1,..., y 2 ) (34) be the fuctios guarateed by Theorem Without loss of geerality, we assume that r(y 1,..., y 2 ) is mootoe icreasig i every coordiate. We ow defie more auxiliary costats. I Theorem 4.12, let costats η ad δ ad fuctios r ad δ be the parameters chose i (32) (34). Theorem 4.12 guaratees iteger costats t = t (4.12) (η, δ, r, δ) ad 0 = (4.12) 0 (η, δ, r, δ). (35)

20 20 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT We set ε = δ, T = t! ad r = r(t,..., t). (36) Let > 0 be divisible by T ad be sufficietly large wheever eeded. cocludes our discussio of the costats. This 6.2. Proof of Lemma 6.1. With the costats above, let G () be a -graph o vertices satisfyig the hypothesis of Lemma 6.1. Let G () = R () B () be ay two-colorig with colors red ad blue. We prove that oe of R () or B () cotais a copy of F () as a sub-hypergraph which is iduced i G (). With costats η, ad δ ad fuctios r ad δ defied above, we apply Theorem 4.12 to the -graph R () to obtai (η, δ(a P ), a P )-equitable ad t-bouded family of partitios P = P( 1, a P ) with respect to which R () is (δ, r(a P ))- regular. Observe, that due to our choice of r i (36) ad the mootoicity thereof, r(a P ) r. (37) We ow cosider the polyads of P. Set 3 { P ˆ ( 1) bad = ˆP( 1) P ˆ( 1) : K ( ˆP } ( 1) ) < / log. Note that the t-boudedess of P gives for sufficietly large { K ( ˆP ( 1) ): ˆP( 1) P ˆ } ( ) ( 1) 1 a1 bad a j) ( j log Set { P ˆ ( 1) reg = ˆP( 1) P ˆ( 1) \ ˆ P ( 1) bad : j=2 c ) 4 ( s R () is (δ,, r(a P ))-regular w.r.t. ˆP ( 1) }. ( ). (38) While P ˆreg ( 1) is defied i terms of the -graph R () oly, the followig fact observes that both R () ad B () are regular w.r.t. every polyad ˆP ( 1) P ˆ( 1) Fact 6.2. ˆP ( 1) ˆ P ( 1) reg = B () is (2δ,, r(a P ))-regular w.r.t. ˆP ( 1). Proof of Fact 6.2. Ideed, for fixed ˆP ( 1) ˆ P reg ( 1), we ow (39) reg. (1) R () is (δ,, r(a P ))-regular w.r.t. ˆP ( 1) (by defiitio of P ˆreg ( 1) ); (2) G () is (ε, d(g () ˆP ( 1) ), r)-regular w.r.t. ˆP( 1), where d(g () ˆP ( 1) ) [ 1 4, 3 4 ] (see (ii ) of Lemma 6.1). 3 We ote that oe could, i fact, show that ˆP( 1) bad =. This would follow from the fact that there are oly a bouded umber (idepedet of ) of polyads ˆP ( 1) ˆP ( 1) ad each of them correspods to a (δ, (1/a 2,..., 1/a 1 ))-regular (/a 1,, 1)-complex. I this situatio, oe ca argue that with δ mi{1/a 1,..., 1/a 1 } we have K ( ˆP ( 1) ) = h (1 ± f(δ)) Q 1 h=2 (1/a h) (/a 1 ) where f(δ) 0 as δ 0. Rather tha maig this precise, however, we chose i our curret proof to use the fact that (sparse) polyads ˆP ( 1) ( 1) ˆP bad ca have oly little ifluece.

21 EXTREMAL HYPERGRAPH PROBLEMS AND THE REGULARITY METHOD 21 As such, it may be directly verified from Defiitio 4.10 that the differece B () = G () \ R () is (ε + δ,, mi{r(a P ), r})-regular w.r.t. ˆP( 1) (with complemetary desity d(b () ˆP ( 1) ) = d(g () ˆP ( 1) ) d(r () ˆP ( 1) )). Recallig ε = δ from (36) ad r(a P ) r from (37), Fact 6.2 follows. We proceed with the first of two easy claims that will prove Lemma 6.1. Claim 6.3. For s = R () (f, f) fixed i (31), there exists S Cross s (P (1) ) so that every K ( ) S has ˆP( 1) (K) P ˆ( 1) reg. Proof of Claim 6.3. Set G () = G () Cross (P (1) ) { K ( ˆP ( 1) ): ˆP( 1) ˆ P reg ( 1) }. (40) Observe that every S K s ( G () ) satisfies the properties required by the claim. As such, it suffices to prove K s ( G () ) > 0. Recall that our hypothesis i Lemma 6.1 assumes that K s (G () ) > c ( s). We show that, i deletig the few edges of G () to obtai G (), we do t destroy all of these cliques. First, we chec that G () \ G () is small. Ideed, sice P is a (η, δ(a P ), a P )- equitable family of partitios, G () \ Cross (P (1) ) ( ) η. (41) Combiig (38) with the fact that R () is (δ, r(a P ))-regular w.r.t. P we have (i view of (39)) that { G () \ K ( ˆP ( 1) ): ˆP( 1) P ˆ } ( reg ( 1) δ + c )( ) 4 ( ) s. (42) Cosequetly, we ifer from (40), (41), ad (42) that ( G () \ G () η + δ + c )( ) 4 ( s ) (32) 3c 4 Now, sice each -tuple of G () \ G () ca belog to at most ( s we see that (43) implies K s ( G () ) K s (G () ) 3c 4 = Ks (G () ) 3c 4 ( s ( ) 1 ( )( ) s s ( ) (i ) c ( ) > 0 s 4 s where we used property (i ) from the hypothesis of Lemma 6.1. ) 1 ( ). (43) ) () cliques K As guarateed by Claim 6.3, fix S Cross s (P (1) ) of size s = R () (f, f) whose every K ( ) S has ˆP( 1) (K) P ˆ( 1) reg. We cotiue with the secod of two easy claims that will prove Lemma 6.1. Claim 6.4. There exists a set F ( S f) such that either or d(r () ˆP ( 1) (K)) 1 8 d(b () ˆP ( 1) (K)) 1 8 s, for every K ( F ), (44) for every K ( F ). (45)