HYPERGRAPHS, QUASI-RANDOMNESS, AND CONDITIONS FOR REGULARITY

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1 HYPERGRAPHS, QUASI-RANDOMNESS, AND CONDITIONS FOR REGULARITY YOSHIHARU KOHAYAKAWA, VOJTĚCH RÖDL, AND JOZEF SKOKAN Dedicated to Professors Vera T. Sós and András Hajnal on the occasion of their 70th birthdays Abstract. Haviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-lie hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their wor turned out to be the so called deviation of a hypergraph. They proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this paper, we consider the concept of discrepancy for -uniform hypergraphs with an arbitrary constant density d (0 < d < 1) and prove that the condition of having asymptotically vanishing discrepancy is equivalent to several other quasi-random properties of H, similar to the ones introduced by Chung and Graham. In particular, we prove that the correct spectrum of the s-vertex subhypergraphs is equivalent to quasi-randomness for any s 2. Our wor may be viewed as a continuation of the wor of Chung and Graham, although our proof techniques are different in certain important parts. 1. Introduction and the main result The rich interplay between the investigation of deterministic combinatorial structures and random combinatorial structures has been an important feature of modern combinatorics. One aspect of this interaction focuses on the study of deterministic structures that mimic the behavior of random ones, from certain specific points of view. In this paper, we are interested in quasi-random hypergraphs, in the sense of Chung and Graham [5, 6]. Haviland and Thomason [9, 10], Chung [4], and Chung and Graham [5, 6] have already established the fundamental results in this area. Babai, Nisan, and Szegedy [3] have implicitly found a connection between communication complexity and what is nown as hypergraph discrepancy, a ey concept, as we shall see, in the study of quasi-random hypergraphs. This connection was explored further by Chung and Tetali [7]. Here, we carry out our investigation very much along the lines of Chung and Graham [5, 6], except that we focus on Date: December 6, The first author was partially supported by FAPESP (Proc. 96/ ), by CNPq (Proc /99 7, /00 0, and /93 1), and by MCT/CNPq (PRONEX project 107/97). The second author was partially supported by NSF grants DMS , DMS and INT The third author was partially supported by NSF grant INT The authors are grateful to the referee for his or her comments.

2 2 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN hypergraphs of arbitrary constant density, maing use of different techniques in certain delicate parts. In the remainder of this introduction, we carefully discuss a result of Chung and Graham [5] and state our main result, Theorem 1.3 below The result of Chung and Graham. We need to start with some definitions. For a set V and an integer 2, let [V ] denote the system of all -element subsets of V. A subset G [V ] is called a -uniform hypergraph. If = 2, we have a graph. We sometimes use the notation G = (V (G), E(G)). If there is no danger of confusion, we shall identify the hypergraphs with their edge sets. In particular, we write H for the number of edges in H. Throughout this paper, the integer is assumed to be a fixed constant. For any l-uniform hypergraph G and l, let K (G) be the set of all -element sets that span a clique K (l) on vertices. We also denote by K (2) the complete -partite -uniform hypergraph whose every partite set contains precisely two vertices. We refer to K (2) as the generalized octahedron, or, simply, the octahedron. We also consider a function µ H : [V ] { 1, 1} such that, for all e [V ], we have { 1, if e H µ H (e) = 1, if e H. Let [] = {1, 2,..., } and let V 2 denote the set of all 2-tuples (v 1, v 2,..., v 2 ), where v i V (1 i 2). Furthermore, let Π () H : V 2 { 1, 1} be given by Π () H (u 1,..., u, v 1,..., v ) = µ H (ε 1,..., ε ), ε=(ε i) i=1 where the product is over all vectors ε = (ε i ) i=1 with ε i {u i, v i } for all i and we understand µ H to be 1 on arguments with repeated entries. The deviation dev(h) of H is defined by dev(h) = 1 m 2 u i,v i V, i [] Π () H (u 1,..., u, v 1,..., v ). Note that the quantity m 2 dev(h) is essentially the difference between the number of 2-tuples that induce an even number of edges and the number of 2-tuples that induce an odd number of edges. For two hypergraphs G and H, we denote by ( H G) the set of all induced subhypergraphs of H that are isomorphic to G. We also write ( H w G) for the number of wea (i.e., not necessarily induced) subhypergraphs of H that are isomorphic to G. Furthermore, we need the notion of the lin of a vertex. Definition 1.1. Let H be a -uniform hypergraph and x V (H). We shall call the ( 1)-uniform hypergraph H(x) = {e \ {x}: e H, x e} the lin of the vertex x in H. For a subset W V (H), we define H(W ) by H(W ) = H(x). x W For simplicity, if W = {x 1,..., x }, we write H(x 1,..., x ).

3 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 3 Observe that if H is -partite, then H(x) is ( 1)-partite for every x V. Furthermore, if = 2, then H(x) may be identified with the set of all vertices connected to x, i.e., H(x) is the neighborhood of x; furthermore, H(x, x ) is the set of all vertices connected to both x and x, i.e., H(x, x ) is the joint neighborhood of x and x. In [5], Chung and Graham proved that if the density of an m-vertex -uniform hypergraph H is 1/2, i.e., H = (1/2 + o(1)) ( ) m, where o(1) 0 as m, then the following statements are equivalent: Q 1 (s): for all -uniform hypergraphs G on s 2 vertices and automorphism group Aut(G), ( ) ( ) H m = (1 + o(1)) 2 (s ) s! G s Aut(G), Q 2 : for all -uniform hypergraphs G on 2 vertices and automorphism group Aut(G), ( ) ( ) H m = (1 + o(1)) 2 (2 ) (2)! G 2 Aut(G), Q 3 : dev(h) = o(1), Q 4 : for almost all choices of x, y V, the ( 1)-uniform hypergraph H(x) H(y), that is, the complement [V ] 1 \ (H(x) H(y)) of the symmetric difference of H(x) and H(y), satisfies Q 2 with replaced by 1, Q 5 : for 1 r < 2 and almost all x, y V, ( ) ( ) H(x, y) m = (1 + o(1)) 2 ( 1) r. r K ( 1) r The equivalence of these properties is understood in the following sense. For two properties involving o(1) terms P = P (o(1)) and P = P (o(1)), the implication P P means that for every ε > 0 there is a δ > 0 so that any -uniform hypergraph H on m vertices satisfying P (δ) must also satisfy P (ε), provided m > M 0 (ε). Chung and Graham [5] stated that it would be profitable to explore quasirandomness extended to simulating random -uniform hypergraphs G p (n) for p 1/2, or, more generally, for p = p(n), especially along the lines carried out so fruitfully by Thomason [12, 13]. Our present aim is to tae the first steps in this direction. In this paper, we concentrate on the case in which p is an arbitrary constant. In certain crucial parts, our methods are different from the ones of Chung and Graham. In fact, it seems to us that the fact that the density of H is 1/2 is essential in certain proofs in [5] (especially those involving the concept of deviation) Discrepancy and the subgraph counting formula. The following concept was proposed by Franl and Rödl and was later used by Chung [4] and Chung and Graham in [5, 6]. For an m-vertex -uniform hypergraph H with vertex set V, we define the density d(h) and the discrepancy disc 1/2 (H) of H as follows: ( ) 1 m d(h) = H and disc 1/2 (H) = 1 m max G [V ] 1 H K (G) H K (G), (1)

4 4 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN where the maximum is taen over all ( 1)-uniform hypergraphs G with vertex set V, and H is the complement [V ] \ H of H. To accommodate arbitrary densities, we extend the latter concept as follows. Definition 1.2. Let H be a -uniform hypergraph with vertex set V with V = m. We define the discrepancy disc(h) of H as follows: disc(h) = 1 m max G [V ] 1 H K (G) d(h) K (G), (2) where the maximum is taen over all ( 1)-uniform hypergraphs G with vertex set V. Observe that if d(h) = 1/2, then disc(h) = (1/2) disc 1/2 (H), so both notions are equivalent. Following some initial considerations by Franl and Rödl, Chung and Graham investigated the relation between discrepancy and deviation. In fact, Chung [4] succeeded in proving the following inequalities closely connecting these quantities: (i) dev(h) < 4 (disc 1/2 (H)) 1/2, (ii) disc 1/2 (H) < (dev(h)) 1/2. For simplicity, we state the inequalities for the density 1/2 case. For the general case, see Section 5 of [4]. Before we proceed, we need to introduce a new concept. If the vertex set of a hypergraph is totally ordered, we say that we have an ordered hypergraph. Given two ordered hypergraphs G and H, where and denote the orderings on the vertex sets of G = G and H = H, we say that a function f : V (G) V (H) is an embedding of ordered hypergraphs if (i) it is an injection, (ii) it respects the orderings, i.e., f(x) f(y) whenever x y, and (iii) f(g) H if and only if g G, where f(g) is the set formed by the images of all the vertices in g. Furthermore, if G = G and H = H, we write (H ) ( ) H = G for the number of such embeddings. We use the same symbol for the orders involved in case this causes no confusion. As our main result, we shall prove the following extension of Chung and Graham s result (in the sense that the density of H is allowed to be different from 1/2). Theorem 1.3. Let H = (V, E) be a -uniform hypergraph of density d on m vertices. Then the following statements are equivalent: P 1 : disc(h) = o(1), P 2 : disc(h(x)) = o(1) and d(h(x)) = (1 + o(1))d for all but o(m) vertices x V and disc(h(x, y)) = o(1) and d(h(x, y)) = (1 + o(1))d 2 for all but o(m 2 ) pairs x, y V, P 3 : disc(h(x, y)) = o(1) and d(h(x, y)) = (1 +o(1))d 2 for all but o(m 2 ) pairs x, y V, P 4 : the number of non-induced copies of K (2) in H is asymptotically minimized among all -uniform hypergraphs of density d; indeed, ( ) w H = (1 + o(1)) m2 K (2) 2! d2, (3) G

5 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 5 P 5 : for every s 2 and all -uniform hypergraphs G on s vertices with e(g) edges and automorphism group Aut(G), ( ) ( ) H m = (1 + o(1)) d e(g) s! (1 d) (s ) e(g) G s Aut(G), P 5 : for every ordering H of H and for every fixed integer s 2, every ordered -uniform hypergraph G on s vertices with e(g) edges is such that ( ) ( ) H m G = (1 + o(1)) d e(g) (1 d) (s ) e(g), s P 6 : for all -uniform hypergraphs G on 2 vertices with e(g) edges and automorphism group Aut(G), ( ) ( ) H m = (1 + o(1)) d e(g) (1 d) (2 ) e(g) (2)! G 2 Aut(G). P 6 : for every ordering H of H, every ordered -uniform hypergraph G on 2 vertices with e(g) edges is such that ( ) ( ) H m G = (1 + o(1)) d e(g) (1 d) (2 ) e(g). 2 The equivalence between properties is understood in the sense of Chung and Graham s approach. Note that, similarly to the case where = 2 (see, e.g., [1, 2]), the equivalence among the above properties may be used to develop a fast algorithm for checing whether a given hypergraph is quasi-random. While it is hard to chec whether disc(h) δ directly from the definition of disc(h), one may chec property P 4 in O(m 2 ) time. This may be further improved using techniques from [1, 11]. Some of the implications in Theorem 1.3 are fairly easy or are by now quite standard. There are, however, two implications that appear to be quite difficult. The proof of Chung and Graham that dev 1/2 (H) = o(1) implies P 5 (the subgraph counting formula ) is based on an approach that has its roots in a seminal paper of Wilson [14]. This beautiful proof seems to mae non-trivial use of the fact that d(h) = 1/2. Our proof of the implication that small discrepancy implies the subgraph counting formula (P 1 P 5) is based on a different technique, which wors well in the arbitrary constant density case (see Section 6). The second implication with a rather technical proof is P 2 P 1. This proof is based on the observation that in -uniform -partite hypergraphs the regularity of lins and pair-lins implies the regularity of the whole hypergraph. For details, we refer the reader to Sections 3.1 and 4. Remar. Let us mae some remars on the asymptotic notation that we shall use. Unless otherwise stated, we understand by o(1) a function approaching zero as the number of vertices of a given hypergraph goes to infinity. We also use x y if x = (1 + o(1))y and x y if x (1 + o(1))y. Finally, we write O 1 (x) for a term y such that y x. 2. Definitions Besides introducing some definitions and notation, our aim in this section is to argue that, for most of the purposes of this paper, we may restrict ourselves to the

6 6 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN case of -partite -uniform hypergraphs. To this end, we first set up a few facts concerning -partite hypergraphs Definitions for partite hypergraphs. For simplicity, we first introduce the term cylinder to mean partite hypergraphs. Definition 2.1. Let l 2 be two integers. We shall refer to any -partite l-uniform hypergraph H = (V 1 V, E) as a -partite l-cylinder or (, l)- cylinder. If l = 1, we shall often write H i for the subhypergraph of H induced on j i V j. Clearly, H = i=1 H i. We shall also denote by K (l) (V 1,..., V ) the complete (, l)-cylinder with vertex partition V 1 V. Definition 2.2. For a (, l)-cylinder H, we shall denote by K j (H), l j, the (, j)-cylinder whose edges are precisely those j-element subsets of V (H) that span cliques of order j in H. Clearly, the quantity K j (H) counts the total number of cliques of order j in H. In the case in which l = 1, the (, j)-cylinder K j (H) is the complete -partite j-uniform hypergraph on H = h H h. When we deal with cylinders, we have to measure density according to their natural vertex partitions. Definition 2.3. Let H be a (, )-cylinder with -partition V = V 1 V. We define the -partite density or simply the density d(h) of H by d(h) = H V 1... V. To be precise, we should have a distinguished piece of notation for the notion of -partite density. However, the context will always mae it clear which notion we mean when we tal about the density of a (, )-cylinder. We should also be careful when we tal about the discrepancy of a cylinder. Definition 2.4. Let H be a (, )-cylinder with vertex set V = V 1 V. We define the discrepancy disc(h) of H as follows: 1 disc(h) = max H K (G) d(h) K (G), (4) V 1... V G [V ] 1 where the maximum is taen over all (, 1)-cylinders G with vertex set V = V 1 V. We now introduce a simple but important concept concerning the regularity of a (, )-cylinder. Definition 2.5. Let H be a (, )-cylinder with -partition V = V 1 V and let δ < α be two positive real numbers. We say that H is (δ, α)-regular if the following condition is satisfied: if G is any (, 1)-cylinder such that then K (G) δ V 1... V, (5) (α δ) K (G) H K (G) (α + δ) K (G). (6) One should observe that the (δ, α)-regularity of a hypergraph H does not imply that H has density α; we may only conclude that the density of a (δ, α)-regular hypergraph is between α δ and α + δ. Moreover, the following simple facts hold. Fact 2.6. Let H be a (δ, α)-regular (, )-cylinder. Then disc(h) 2δ.

7 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 7 Fact 2.7. Suppose H is a (, )-cylinder with -partition V = V 1 V. Put α = d(h) and assume that disc(h) δ. Then H is (δ 1/2, α)-regular The -partite reduction. Suppose H is a -uniform hypergraph and let H be one of its -partite spanning subhypergraphs. In this section, we try to relate the deviation and the discrepancy of H to those of H. Definition 2.8. Let H = (V, E) be a -uniform hypergraph with m vertices and let P = (V i ) 1 be a partition of the vertex set V. We denote by H P the (, )-cylinder consisting of the edges h H satisfying h V i = 1 for all i []. The following proposition holds. Proposition 2.9. For any partition P = (V i ) 1 of V, we have (i) disc(h) d(h P ) d(h) V 1... V /m, and (ii) disc(h P ) 2 disc(h)m / V 1... V. Proof. Let P = (V i ) 1 be any partition of V. Observe that H P consists precisely of the vertex sets of those copies of K ( 1) in K = K ( 1) (V 1,..., V ) which are also edges in H; that is, H P = H K (K). Since K (K) = K (K ( 1) (V 1,..., V )) = V 1... V, this implies the first part of the proposition by taing G = K in (2). On the other hand, let G 0 [V ] 1 be a (, 1)-cylinder for which the maximum is attained in (4), the definition of disc(h P ). Observe that H P K (G 0 ) = H K (G 0 ). Then 1 disc(h P ) = H P K (G 0 ) d(h P ) K (G 0 ) V 1... V 1 H K (G 0 ) d(h) K (G 0 ) V 1... V 1 + d(h) K (G 0 ) d(h P ) K (G 0 ) V 1... V m V 1... V disc(h) + 1 V 1... V K (G 0 ) d(h) d(h P ) m V 1... V disc(h) + d(h) d(h P) 2m V 1... V disc(h), where in the last inequality we used (i). We shall also need the following fact, which follows easily from, say, Chebyshev s inequality. Fact Let H = (V, E) be an m-vertex -uniform hypergraph. o(1)) m partitions P = (V i ) 1 of V satisfy (i) V i = (1 + o(1))m/ for all i [], (ii) H P = (1 + o(1))(!/ ) H, and (iii) d(h P ) = (1 + o(1))d(h), where o(1) 0 as H /m 1. Then (1 An immediate consequence of the previous proposition and Fact 2.10 is the following.

8 8 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN Claim If disc(h) = o(1), then disc(h P ) = o(1) for (1 o(1)) m partitions P = (V i ) 1 of V. With some more effort, one may prove a converse to Claim Claim Suppose there exists a positive real number γ > 0 such that disc(h P ) = o(1) for γ m partitions P = (V i ) 1 of V. Then disc(h) = o(1). Proof. Let S be a set of partitions P for which disc(h P ) = o(1) and S γ m. Suppose disc(h) δ for some fixed δ > 0, and let G 0 be a ( 1)-uniform hypergraph for which the maximum is attained in (2), the definition of disc(h). Let P = (V i ) 1 S be a partition satisfying the conclusion of Fact 2.10 with respect to H, H K (G 0 ), and K (G 0 ). Such a partition must exist since γ m +(1 o(1)) m > m. Observe that, then, and H P K (G 0 ) = (H K (G 0 )) P = (1 + o(1))! H K (G 0 ), K (G 0 K ( 1) (V 1,..., V )) = (K (G 0 )) P = (1 + o(1))! K (G 0 ), and, from (iii) of Fact 2.10, d(h P ) = (1 + o(1))d(h). For convenience, put K = K ( 1) (V 1,..., V ). We use an approach similar to the one in Proposition 2.9 to get disc(h) = 1 H K m (G 0 ) d(h) K (G 0 ) = 1 m (1 + o(1))! H P K (G 0 ) (1 + o(1))d(h P )! K (G 0 K) 1 H m P K (K G 0 ) d(h P ) K (G 0 K)! + 1 m! o(1) ( H P K (K G 0 ) + d(h P ) K (G 0 K) ) (1 + o(1)) 1! disc(h P) + 1 ( ) m m! o(1)2 2! disc(h P) + 2 (!) 2 o(1). Since by our assumptions disc(h P ) = o(1), we immediately obtain that disc(h) < δ for large enough m, which is a contradiction. We now state the -partite version of a part of our main result, Theorem 1.3. Theorem Suppose V = V 1 V, V 1 = = V = n, and let H = (V, E) be a (, )-cylinder with H = dn. Then the following four conditions are equivalent: C 1 : H is (o(1), d)-regular; C 2 : H(x) is (o(1), d)-regular for all but o(n) vertices x V and H(x, y) is (o(1), d 2 )-regular for all but o(n 2 ) pairs x, y V ; C 3 : H(x, y) is (o(1), d 2 )-regular for all but o(n 2 ) pairs x, y V ; C 4 : the number of copies of K (2) in H is asymptotically minimized among all such (, )-cylinders of density d, and equals (1 + o(1))n 2 d 2 /2.

9 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 9 Remar. The simplifying condition V 1 = = V = n has the sole purpose of maing the proof more readable and transparent. The immediate generalization of Theorem 2.13 for V 1,..., V of arbitrary sizes holds. The proof of Theorem 2.13 will be given in Sections 4 and The derivation of the general case In this part, we prove Theorem 1.3. We divide this proof into five sections. In Section 3.1, we show the equivalence of properties P 1, P 2, and P 3. The proof of P 4 P 1 is in Section 3.2. Both sections use Theorem 2.13 as the main tool. In Section 3.3, we prove P 1 P 5 using the subhypergraph counting formula from Section 6. Then we show that P 5 P 5 P 6 and P 5 P 6 P 6 (see Section 3.4). Finally, we prove P 6 P 4 in Section 3.5. The flow of the whole proof is described in the following diagram. P 2 P 3 P 1 P 1 P 6 P 5 P 4 P 6 P Proof of P 1 P 2 P 3. We are now ready to show that, in the first part of the proof of Theorem 1.3, we may assume the hypergraph H to be -partite. To be more precise, we show that the equivalence P 1 P 2 P 3 in Theorem 1.3 follows from Theorem We shall illustrate this on P 1 P 2 ; the other implications are handled similarly. In fact, we shall be somewhat setchy; we shall only indicate the double counting argument that gives this result. Suppose that we have a -uniform hypergraph H with density d such that disc(h) = o(1). From Fact 2.10 and Claim 2.11 we now that for all but o(1) m partitions P of V we have disc(h P ) = o(1) and d(h P ) = (1 + o(1))d. For every partition P, denote by X(P) the set of all vertices x V such that either disc(h P (x)) o(1) or disc(h P ) o(1). From Theorem 2.13 and Facts 2.6 and 2.7, we now that for all but o(1) m partitions P, we have X(P) = o(m). For the remaining o(1) m partitions P, we use X(P) m. For a vertex x V, we define P(x) to be the collection of all partitions P for which x X(P). One can easily see that X(P) = P(x). P x V Let S be the set of vertices x V for which P(x) > (1/2) m. Then o(1) m m + o(m) m P X(P) = x V P(x) > 1 2 S m, hence S = o(m). This means that, for almost all vertices x V, we have disc(h(x) P ) = disc(h P (x)) = o(1) for at least (1/2) m partitions P. By Claim 2.12, it follows that disc(h(x)) = o(1) for all but o(m) vertices x V. We proceed similarly in order to show that disc(h(x, y)) = o(1) for all but o(m 2 ) pairs x, y V.

10 10 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN 3.2. The minimization of the number of octahedra. The aim of this section is to show that property P 1 can be derived from property P 4 using the equivalence of the -partite properties C 1 and C 4. We start with the following lemma. Lemma 3.1. Suppose a -uniform hypergraph H on m vertices with density d = d(h) is such that ( ) w H m2 K (2) 2! d2. (7) Then, for almost all partitions P of V = V (H), ( ) w HP n2 K (2) 2 d2 (8) holds for n = m/. Proof. Put ( ) w X(P) = HP. K (2) We consider X as a r.v. on the uniform probability space of all partitions P of the vertex set V of H. Clearly, we may write X as a sum of 0 1 indicator random variables as follows: X = X K, K where the sum is over all K H with K = K (2) and X K (P) = 1 if and only if K H P. Note that P(X K = 1) = P(K H P ) =! m 2 / m. Therefore, using (7), we have E(X) = E(X K ) = P(X K = 1) m2! m 2 2! d2 m n2 2 d2. (9) K K We now invoe a lemma that will be proved in Section 5.1. Indeed, Claim 5.2 states that ( ) w G n2 K (2) 2 d2 (10) for all (, )-cylinders G of density d with n vertices in each of its vertex classes. Comparing (9) and (10), we deduce that the expectation of X is asymptotically equal to min X, and hence P ) (X (1 + η) n2 2 d2 = o(1) for any fixed η > 0. This completes the proof of Lemma 3.1. We turn to the proof of the implication P 4 P 1. Proof of P 4 P 1. Let H be a -uniform hypergraph on m vertices such that ( ) w H m2 K (2) 2! d2, (11) where d is the density d(h) of H. Lemma 3.1 then implies that almost all vertex partitions P = (V i ) 1 are such that ( ) w HP n2 K (2) 2 d2, (12)

11 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 11 where n = m/. The implication C 4 C 1 of Theorem 2.13 gives that, then, the (, )-cylinder H P is (o(1), d(h P ))-regular for almost all P. We now use Fact 2.6 to conclude that H P satisfies disc(h P ) = o(1) for a.a. P. We may then apply Claim 2.12 with, say, γ = 1/2, to deduce that disc(h) = o(1); in other words, property P 1 holds for H Proof of P 1 P 5. In this section, we prove that any -uniform hypergraph H with disc(h) = o(1) must be such that any -uniform hypergraph G on s vertices must occur as an induced subhypergraph of H as many times as one would expect if H were a truly random hypergraph with density d. Our proof will be based on a certain technical result, the subhypergraph counting formula, which will be proved in Section 6. Proof of P 1 P 5. We need to show that for every given integer s 2, real number ε > 0, and density d (0, 1), there exists a real number δ > 0 such that property P 1 (δ) (i.e., property P 1 with o(1) replaced by δ) implies property P 5 (ε, s) (i.e., property P 5 with given s and o(1) replaced by O 1(ε)). Let δ 0 = δ 0 (d, ε) be the positive real number determined by Corollary 6.13 and d 0 = min{d, 1 d}. Choose δ > 0 of the form 1/t 2, where t N, satisfying δ = 1 ( ) 4 t 2 δ0, (13a) 2 and δ 1/2 = 1 ) t εd(s 0 100s 2 1 s. (13b) Let m m(d, ε) be an integer divisible by t and set n = m/t. Suppose that H = H and G = G are two ordered -uniform hypergraphs such that V (H) = {v 1 v 2 v m }, V (G) = {w 1 w 2 w s }, d(h) = d, and disc(h) δ. For every i [t] set V i = {v j+n(i 1) : j [n]} and note that V (H) = t i=1 V i is a partition of V (H). An s-tuple {u 1,..., u s } is crossing, or transversal, if {u1,..., u s } V i 1 for all i [t]. Note that the number of non-crossing s- tuples is bounded from above by ( )( ) n m 2 t 2 s 2 = 1 2 m ( m t 1 ) ( m 2 s 2 = ) (s)2 m(m t) t(m) 2 (s) 2 ( s 2 )( ) m 2 (m)2 1 s 2 (s) 2 t Since the number of crossing s-tuples is ( t s) n s, we have the following fact. Fact 3.2. ( ) ( m s t ) s n s ( s m ) 2)( s /t. ( s 2 )( m s For I [t], put H I = H [ i I V i] and observe that HI is an ( I, )-cylinder. One can mimic the proof of Proposition 2.9 and obtain the following fact. Fact 3.3. For every I [t], we have d(h I ) = d + O 1 (δ 1/2 ) and disc(h I ) 2δ t = 2δ 1/2. Consequently, owing to Fact 2.7, the cylinder H I is (2δ 1/4, d)-regular. ).

12 12 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN Thus, the (s, )-cylinder H I satisfies the assumptions of Corollary 6.13 for every I [t] s. Therefore, there exist (1 + O 1 (ε))d e(g) (1 d) (s ) e(g) n s transversal copies of G in each H I (I [t] s ). Let I = {i 1 < i 2 < < i s } [t] s and consider one transversal copy of G in H on vertices u 1 u 2 u s, where u j V ij for every j [s]. Then the mapping ϕ: V (G) V (H) defined by ϕ(w i ) = u i (i [s]) is an injection preserving order, and preserving edges and non-edges of G; thus, ϕ ( ) H G. In view of the previous paragraph, we have ( ) t (1 + O 1 (ε))d e(g) (1 d) (s ) e(g) n s (14) s such mappings. On the other hand, let ϕ ( H G ). The s-tuple {ϕ(w 1) ϕ(w 2 ) ϕ(w s )} is either crossing or not. In the first case, this s-tuple induces a transversal copy of G in H, and, therefore, equation (14) yields the number of mappings ϕ for which this case occurs. In the second case, {ϕ(w 1 ) ϕ(w 2 ) ϕ(w s )} is not crossing. By Fact 3.2, there are at most ( s m ) 2)( s /t mappings ϕ yielding this case. Combining these two cases together implies that ( ) H G = (1 + O 1(ε))d e(g) (1 d) (s ) e(g) n s ( ) ( t 1 + O 1 s t ( s 2 )( m s )). To complete the proof, it suffices to show that ( ) ( ( )( )) t 1 s m (1 + O 1 (ε))d e(g) (1 d) (s ) e(g) n s + O 1 s t 2 s ( ) m = (1 + O 1 (2ε)) d e(g) (1 d) (s ) e(g). (15) s Owing to (13b), we have ( )( ) 1 s m t 2 s and t s < ε 100 ( ) m d e(g) (1 d) (s ) e(g), s (16a) ( ) s t s 1 > t s ε ts. (16b) Since (16a) holds, (15) follows from the following inequality ( ) ( ) ( ) m t m (1 ε/2) n s (1 + ε/2). s s s While the right-hand side of this inequality is immediate, the left-hand side is a consequence of (16b) Proof of P 5 P 6 P 6 and P 5 P 5 P 6. Implications P 5 P 6 and P 5 P 6 are trivial since P 6 (respectively, P 6) is a special case of P 5 (respectively, P 5). Moreover, P 6 P 6 is a special case of P 5 P 5, therefore, it suffices to prove that P 5 P 5.

13 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 13 G) w inj ) Proof of P 5 P 5. Given two hypergraphs G and H, let us denote by ( ) H G the inj set of injections ϕ: V (G) V (H) such that ϕ(g) H if and only if g G. Moreover, we write ( H for the set of such injections such that ϕ(g) H whenever g G. Thus, ( H G is the set of embeddings ϕ of G into H such that ϕ(v (G)) inj induces an isomorphic copy of G in H, whereas ( H w G) is the set of embeddings ϕ inj such that ϕ(v (G)) induces a superhypergraph of G in H. If G has automorphism group Aut(G), it is easy to verify that we have ( ) H ( ) G inj = H Aut(G), (17) G and similarly for ( ) H w G and ( H w. inj G) Suppose now that H is an ordered hypergraph with ordering. Then, ( ) ( H H G inj = G ), (18) where the union ranges over the set of all total orderings of V (G). Furthermore, a moment s thought shows that the union in (18) is a disjoint union. Hence ( ) H G inj = ( ) H. (19) G Since P 5 holds, we have ( ) H = (1 + o(1)) G ( m s ) d e(g) (1 d) (s ) e(g) for every total ordering of V (G). Since there exist s! total orderings of V (G), combining (17) and (19) yields P Minimization of octahedra from subhypergraph counting. We now prove that property P 6 (which concerns a certain subhypergraph counting formula for induced subhypergraphs) implies property P 4 (which concerns the number of (wea) subhypergraphs isomorphic to octahedra). The proof will have two parts. In the first part we shall show that, for every hypergraph H with density d, the number of copies of K (2) in H is bounded from below by (1 + o(1)) m2 2! d2 (see Lemma 3.5). In the second part, we shall prove that P 6 implies the asymptotic equality (3) given in property P 4 of Theorem 1.3. We start with the following lemma. Lemma 3.4. Suppose the -uniform hypergraph H with m vertices and with density d = d(h) is such that H P satisfies ( ) w HP n2 K (2) 2 d2 (20) for almost all partitions P = (V i ) 1 of the vertex set V = V (H), where n = m/. Then ( ) w H m2 K (2) 2! d2. (21)

14 14 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN Proof. This lemma follows easily from a double counting argument. Let us consider the family of pairs (K, P) such that (i) K (2) = K H, (ii) P = (V i ) 1 is such that (20) holds and V i n = m/ (1 i ), and, finally, (iii) K H P. On the one hand, the number N of such pairs (K, P) is { K : (i) and (iii) hold }, (22) P where the sum is over all P for which (ii) holds. Thus, because of our assumption on H and Fact 2.10, we have that On the other hand, we have that N = K N m n2 2 d2. (23) { P : (ii) and (iii) hold } ( ) w H! m 2, (24) K (2) where the sum is over all K that satisfy (i). Above, we again made use of Fact 2.10 to estimate the number of relevant partitions P for each fixed K. Comparing (23) and (24), we deduce (21). The proof of the lower bound on the number of K (2) in H is straightforward now. Lemma 3.5. For any m-vertex -uniform hypergraph H with density d = d(h), ( ) w H m2 K (2) 2! d2. (25) Proof. We now that all but o(1) m partitions P satisfies V i = n m/ and d(h P ) d (see Fact 2.10). By Claim 5.2 we now that ( ) HP n2 K (2) 2 d2, and, therefore, by Lemma 3.4, (25) holds. Proof of P 6 P 4. Let H be a -uniform hypergraph on m vertices such that, for any -uniform hypergraph G on 2 vertices, we have ( ) ( ) H m d e(g) (1 d) (2 ) e(g) (2)! G 2 Aut(G), (26) where d = d(h) is the density of H and Aut(G) is the automorphism group of G. From (17) it follows that, for any such G, we have ( ) H G (m) 2d e(g) (1 d) (2 ) e(g), (27) inj

15 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 15 where, as usual, (a) b = a(a 1)... (a b + 1). We are interested in estimating ( H w G). Clearly, inj ( ) w H = ( ) H G inj G, (28) G inj where the union ranges over all -uniform hypergraphs G with the same vertex set as G and G G. Furthermore, a moment s thought shows that the union in (28) is a disjoint union. Hence ( ) w H G = ( ) H G G Thus, inj inj G (m) 2 d e(g ) (1 d) (2 ) e(g ) = (m) 2 d e(g)+t (1 d) (2 ) e(g) t t 0 e(g )=e(g)+t = (( 2 ) ) e(g) (m) 2 d e(g)+t (1 d) (2 ) e(g) t t t 0 = (m) 2 d (( 2 ) ) e(g) e(g) d t (1 d) (2 ) e(g) t t t 0 = (m) 2 d e(g). ( H K (2) ) w inj (m) 2d 2. It now suffices to recall the analogue of (17) for wea subhypergraphs to conclude the proof of P 4, since Aut(K (2)) =!2. 4. Proof of C 1 C 2 In this section, we shall prove the equivalence of conditions C 1 and C 2 in Theorem We start with a fairly standard proof of C 1 C 2 (see Section 4.1), and then, in Section 4.2, we prove the converse C 2 C Proof of C 1 C 2. The proof follows from the two claims below. Claim 4.1. Suppose 0 < ε 1/2 < d, V = V 1 V, V 1 = = V = n, and let H = (V, E) be a (ε, d)-regular (, )-cylinder. Then for all but at most 2ε 1/2 n vertices x V, the lin H(x) is (ε 1/2, d)-regular. Proof. Let X be the set of all vertices x V with the following property: there exists a ( 1, 2)-cylinder F x with ( 1)-partition V 1 V 1 such that K 1 (F x ) ε 1/2 n 1, (29) but H(x) K 1 (F x ) < ( d ε 1/2) K 1 (F x ). (30)

16 16 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN We also define X + to be the set of all vertices x V satisfying (29) for which we have ( H(x) K 1 (F x ) d + ε 1/2) K 1 (F x ). Suppose that X > ε 1/2 n and define a (, 1)-cylinder G by G = K ( 1) 1 (V 1,..., V 1 ) {e {x}: e F x } x X Observe that K (G) = x X K 1 (F x ) X ε 1/2 n 1 εn, and, therefore, by the regularity of H, H K (G) (d ε) K (G) = (d ε) x X K 1 (F x ). On the other hand, from (30) we obtain H K (G) = H(x) K 1 (F x ) < (d ε 1/2) K 1 (F x ), x X x X which is a contradiction. Hence X ε 1/2 n. Similarly we obtain X + ε 1/2 n. Claim 4.2. Suppose 0 < ε < 1/16, 2ε 1/2 < d, V = V 1 V, V 1 = = V = n, and let H = (V, E) be a (ε, d)-regular (, )-cylinder. Then, H(x, y) is (ε 1/4, d 2 )-regular for all but at most 4ε 1/4 n 2 pairs of vertices x, y V. Proof. From the previous claim we now that there are at most 2ε 1/2 n vertices x in V with (ε 1/2, d)-irregular lin H(x). These vertices form at most 2ε 1/2 n 2 pairs and we shall exclude them from further considerations. For a vertex x V denote by Yx the set of all vertices y V with the following property: there exists a ( 1, 2)-cylinder F y with ( 1)-partition V 1 V 1 such that K 1 (F y ) ε 1/4 n 1, (31) but H(x, y) K 1 (F y ) < ( d 2 ε 1/4) K 1 (F y ). (32) We also denote by Y x + the set of all vertices y V for which there is a ( 1, 2)- cylinder F y that satisfies (31), but ( H(x, y) K 1 (F y ) > d 2 + ε 1/4) K 1 (F y ). Suppose there exists a vertex x V with (ε 1/2, d)-regular lin H(x) for which Yx ε 1/4 n. Define a (, 1)-cylinder G by G = H(x) {e {y}: e F y }. Then y Y x K (G) = H(x) K 1 (F y ). y Y x

17 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 17 Note that (31) together with the (ε 1/2, d)-regularity of H(x) implies that ( H(x) K 1 (F y ) d ε 1/2) K 1 (F y ) for all y Yx. Hence K (G) ( d ε 1/2) K 1 (F y ) y Y x ( d ε 1/2) ( Yx ε 1/4 n 1 d ε 1/2) ε 1/2 n εn. (33) By the (ε, d)-regularity of H and (33), we have H K (G) (d ε) K (G) (d ε) (d ε 1/2) K 1 (F y ). (34) y Y x On the other hand, the size of H K (G) can be bounded from above using (32) as follows: H K (G) = H(x, y) K 1 (F y ) < (d 2 ε 1/4) K 1 (F y ). (35) y Y x Comparing (34) and (35), we get (d ε) ( d ε 1/2) < d 2 ε 1/4, y Y x which implies ε 1/4 < ( ε + ε 1/2) d < ε + ε 1/2, which is not true for ε < 1/16. Hence we have Yx ε 1/4 n. We also obtain Y x + ε 1/4 n in exactly the same way. Consequently, the number of bad pairs is bounded by 2ε 1/2 n 2 +2ε 1/4 n n 4ε 1/4 n Proof of C 2 C 1. The objective of this section is to prove the following theorem. Theorem 4.3. For every δ > 0 and d > 0 there exist ε > 0 and n 0 N such that the following holds. If H is a ( + 1, + 1)-cylinder with ( + 1)-partition V 1 V +1 such that (i) V i = n n 0 for all i [ + 1], (ii) H = dn +1, (iii) H(x) is (ε, d)-regular for all but at most εn vertices x V +1, and (iv) H(x, y) is (ε, d 2 )-regular for all but at most εn 2 pairs x, y V +1, then H is (δ, d)-regular. Remar. Here, we wor with ( + 1, + 1)-cylinders to simplify the notation. With this choice, we shall encounter ( + 1)-, -, and ( 1)-uniform hypergraphs. Proof. Let H be a ( + 1, + 1)-cylinder satisfying assumptions (i) (iv). We shall assume that ε = (δ/4) 32 < δ < d 1. (36) Suppose that H is not (δ, d)-regular, i.e., Definition 2.5 fails. Without loss of generality (by taing complements) we may assume that the second inequality in

18 18 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN (6) is not true, therefore, there exists a ( +1, )-cylinder G = +1 i=1 G i with ( +1)- partition V 1 V +1 such that K +1 (G) δn +1 (37) but H K +1 (G) > d + δ. (38) K +1 (G) We now wor on this witness G for the irregularity of H. Fact 4.4. There exist subcylinders G i G i (i []) such that G = G 1 G G +1 satisfies the following four conditions: (1) K (G (x)) = or K (G (x)) εn for all x V +1, (2) H(x) is (ε, d)-regular for all x V +1 with K (G (x)) εn, (3) K +1 (G ) (δ/2)n +1, (4) H K +1 (G ) (d + δ/2) K +1 (G ). Proof. For every vertex x V +1 with K (G(x)) < εn or with (ε, d)-irregular lin H(x), delete all edges in G that contain x. Notice that this operation does not remove any edge from G +1 and produces a subhypergraph G = G 1 G G +1 that satisfies conditions (1) and (2). Moreover, every removal reduces the size of K +1 (G) (and H K +1 (G)) by at most εn if K (G(x)) < εn or by at most n if H(x) is (ε, d)-irregular. Since there are at most εn vertices x with (ε, d)-irregular lin, we obtain that K +1 (G ) K +1 (G) n εn εn n, and H K +1 (G ) H K +1 (G) n εn εn n, The first inequality together with assumption (37) shows that K +1 (G ) K +1 (G) 2εn +1 δn +1 2(δ/4) 32 n +1 (δ/2)n +1. Similarly, the second inequality, (37), and (38) yield H K +1 (G ) H K +1 (G) 2εn +1 (d + δ) K +1 (G) 2(δ/4) 32 n +1 and the proof is complete. (d + δ/2) K +1 (G) + (δ/2)δn +1 2(δ/4) 32 n +1 (d + δ/2) K +1 (G ), We have to wor on G further to obtain a witness with more structure. We shall need the following definition. Definition 4.5. For each e G +1 define two parameters g(e) and h(e) by g(e) = {x V+1 : {x} e K +1 (G )}, h(e) = {x V+1 : {x} e H K +1 (G )}. Fact 4.6. Put δ = δ 2 /16. Then there exists a subcylinder G +1 G +1 such that G = G 1 G G +1, where the G i (i []) are taen from Fact 4.4, satisfies the following five conditions: (1) K (G (x)) = or K (G (x)) εn for all x V +1, (2) H(x) is (ε, d)-regular for all x V +1 with K (G (x)) εn, (3) K +1 (G ) δ n +1,

19 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 19 (4) h(e)/g(e) d + δ/4 for all e G +1, (5) g(e) δ n for all e G +1. Proof. We decompose G +1 into two subcylinders G+1 G +1, where { G+1 = e G +1 : h(e) g(e) d + δ } 4 and G+1 = G +1 \G+1. Let G be as in Fact 4.4. Observe that g(e) = K +1 (G Fact 4.4 (3) ) (δ/2)n +1 e G +1 and e G +1 h(e) = H K +1 (G ) Fact 4.4 (4) (d + δ/2) K +1 (G ). Then for G = G 1 G G +1 we have K +1 (G ) = g(e) h(e) = h(e) h(e) e G+1 e G+1 e G +1 e G+1 ( > d + δ ) ( K +1 (G ) d + δ ) g(e) 2 4 e G +1 ( = d + δ ) ( K +1 (G ) d + δ ) K +1 (G ) = δ K +1(G ) δ2 8 n+1. Note that at least (δ 2 /16)n edges e of G +1 must have g(e) (δ2 /16)n, otherwise we would have K +1 (G ) < δ2 16 n n + n δ2 16 n = δ2 8 n+1, which would be a contradiction. Remove all edges e with g(e) < (δ 2 /16)n from G+1 and obtain G +1. Then, G +1 satisfies condition (4) because of the definition of G+1 G +1. It satisfies condition (5) because all edges e with g(e) < (δ2 /16)n have been removed, and G also satisfies condition (3) because K +1 (G ) K +1 (G ) n δ2 16 n δ2 8 n+1 δ2 16 n+1 = δ2 16 n+1. Finally, G must satisfy (1) and (2) because we did not change any of G i (i []). Before we come bac to the proof of Theorem 4.3, we state an auxiliary lemma. Let 0 < α 1 and 0 < µ < 1 be given. Let G be a bipartite graph with vertex classes X 1 X 2 and let H be a subgraph of G. We call an ordered pair of vertices (x, y) X 1 X 1 good if and We also call a pair bad if it is not good. The auxiliary lemma is as follows. H(x) G(y) = α(1 + O 1 (µ)) G(x, y) (39) H(x, y) = α 2 (1 + O 1 (µ)) G(x, y). (40)

20 20 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN Lemma 4.7 (Dementieva, Haxell, Nagle, and Rödl [8]). Let 0 < α 1 and 0 < µ < 1 be given. Suppose the bipartite graphs G and H are such that µα 2 G(x, y) < (1 α) 2 + α 2 G(z) 2. (41) bad (x,y) X 1 X 1 z X 2 Then z X 2 ( H(z) α G(z) ) 2 5µα 2 The following corollary of the above lemma holds. z X 2 G(z) 2. (42) Corollary 4.8. Let 0 < α 1, 0 < µ 1, and ν > 0 be given. Let G and H be bipartite graphs as in Lemma 4.7. Denote by W the set of all vertices z X 2 such that H(z) (α + ν) G(z). Then G(z) ( 5µ/ν 2) 1/2 X1 X 2. (43) z W Proof. Clearly, 5µα 2 z X 2 G(z) 2 5µα 2 X 1 2 X 2 5µ X 1 2 X 2. On the other hand, ( H(z) α G(z) ) 2 α G(z) ) z X 2 z W( H(z) 2 ν 2 G(z) 2 z W ( ) 2 ( 2 ν2 G(z) G(z) ) ν2. W X 2 z W Finally, using inequality (42), we get (43). We now turn bac to the proof of Theorem 4.3. We define two auxiliary bipartite graphs G = (X 1 X 2, E(G)) and H = (X 1 X 2, E(H)) in the following way: X 1 = V +1, X 2 = V 1 X, E(G) = {(x, e): e K (G (x))}, E(H) = {(x, e): e H(x) K (G (x))}. Notice that X 1 = n and X 2 = n. Then G(x) = G(e) G(e) = K +1 (G ) δ n +1 = δ X 1 X 2. x X 1 e X 2 e G +1 (45) Now we prove that the graphs G and H defined by (44) satisfy the assumptions of Corollary 4.8 with α = d, µ = ε 1/2 and ν = δ/4. Indeed, observe that if K (G (x, y)) εn and H(x) is (ε, d)-regular, then H(x) K (G (x, y)) = (d + O 1 (ε)) K (G (x, y)), i.e., H(x) G(y) = d(1 + O 1 (ε 1/2 )) G(x, y), and if K (G (x, y)) εn and H(x, y) is (ε, d 2 )-regular, then H(x, y) K (G (x, y)) = (d 2 + O 1 (ε)) K (G (x, y)), i.e., H(x, y) = d 2 (1 + O 1 (ε 1/2 )) G(x, y). z W (44)

21 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 21 Denote by I 1 the set of all pairs (x, y) such that G(x, y) < ε X 2, by I 2 the set of all pairs (x, y) such that H(x, y) is (ε, d 2 )-irregular, and by I 3 the set of all pairs (x, y) such that H(x) is (ε, d)-irregular. Both observations above imply that every pair (x, y) X 1 X 1 such that H(x) is (ε, d)-regular, H(x, y) is (ε, d 2 )-regular, and G(x, y) ε X 2 is good. In other words, the set of bad pairs is a subset of I 1 I 2 I 3 and, therefore, G(x, y) G(x, y) + G(x, y) + G(x, y). bad (x,y) (x,y) I 1 (x,y) I 2 (x,y) I 3 One can easily see that I 1 X 1 2, I 2 ε X 1 2, I 3 ε X 1 X 1, and G(x, y) ε X 2 I 1 ε X 1 2 X 2, (x,y) I 1 G(x, y) X 2 I 2 ε X 1 2 X 2, (x,y) I 2 G(x, y) X 2 I 3 ε X 1 2 X 2. (x,y) I 3 Consequently, G(x, y) 3ε X 1 2 X 2. bad (x,y) Suppose that condition (41) of Lemma 4.7 is not satisfied, i.e., bad (x,y) G(x, y) d 2 ε 1/2 (1 d) 2 + d 2 z X 2 G(z) 2. Since z X 2 G(z) 2 ( z X 2 G(z) ) 2 / X2, we get ( ) 2 d 2 ε 1/2 (1 d) 2 + d 2 G(z) 3ε X 1 2 X 2 2. z X 2 In other words, z X 2 G(z) ( ( 3ε (1 d) 2 + d 2) ) 1/2 X 1 X 2. d 2 ε 1/2 On the other hand, we now (see (45)) that z X 2 G(z) (δ 2 /16) X 1 X 2, and, therefore, comparing both inequalities yields ( ( δ 2 3ε (1 d) d 2) ) 1/2. d 2 ε 1/2 This is a contradiction since 3ε ( (1 d) 2 + d 2) /(d 2 ε 1/2 ) ε 1/2 /d 2 = (δ/4) 16 /d 2 (δ/4) 14 < δ 4 /256. Thus, G and H also satisfy condition (41) of Lemma 4.7. Set W = G +1. Then, because of property (3), clearly W (δ2 /16)n +1 /n = (δ 2 /16)n = (δ 2 /16) X 2, and for every e W we have H(e) (d + δ/4) G(e). We apply Corollary 4.8 and obtain that ( ) 80ε 1/2 1/2 G(z) X 1 X 2. z W δ 2

22 22 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN On the other hand, since G(e) (δ 2 /16) X 1 for every e W, we get z W G(z) W δ2 16 X 1 δ4 256 X 1 X 2. This is a contradiction because ( ) 80ε 1/2 1/2 ( ) 7 δ δ 2 4 < δ Proof of C 2 C 3 C 4 Sections 5.1, 5.2, and 5.3 are devoted to the proofs of C 3 C 4, C 4 C 3, and C 3 C 2 (note that implication C 2 C 3 trivially holds). In these sections, we shall be setchy in places because the arguments are standard or somewhat repetitive Proof of C 3 C 4. We start with a standard cherry counting lemma for bipartite graphs (a cherry is a path of length 2). Claim 5.1. Let G = (X Y, E) be a bipartite graph with X = n, Y = m, and assume that E = dmn. For x, x X, put d x,x = G(x, x ) /m. Then ( ) dn {dx,x : x, x X, x x }. 2 Proof. Observe first that and G(x) = E = G(y) x X y Y X G(x, x x x ) = ( ) G(y), 2 y Y where we write X x x for the sum over all pairs {x, x } of distinct vertices from X. Then m X x x d x,x = X G(x, x x x ) = ( ) ( G(y) m 1 y Y m G(y) ) ( ) dn = m, y Y where, naturally, we used the convexity of ( x 2). Claim 5.2. Let H = (V 1 V, E) be a (, )-cylinder with E = dn, where V i = n for all i []. Then the number of copies of K (2) in H is bounded from below by (1 + o(1))(n 2 /2 )d 2. Proof. We proceed by induction on. For = 2, the statement follows from the previous claim and the Cauchy Schwarz inequality. Indeed, let G = (X Y, E) be a bipartite graph with X = Y = n, and assume that E = dn 2. For x, x X,

23 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 23 put d x,x = G(x, x ) /n. Here and below we use the notation X x x introduced in the proof of Claim 5.1. Then the number of copies of K 2 (2) in G is given by ( ) G = ( X G(x, x ) ) = ( ) X dx,x n. K 2 (2) x x 2 x x 2 By the Cauchy Schwarz inequality and Claim 5.1, we have ( ) G = ( ) ( ) X dx,x n n X = (1 + o(1)) K 2 (2) x x 2 2 ( X ) 2 ( dn (1 + o(1)) d x x x,x (1 + o(1)) 2 x x d2 x,x ) 2 = (1 + o(1)) n4 4 d4. We now proceed to the induction step. Suppose 3, suppose that the claim is true for 1, and let H = (V 1 V, E) be a (, )-cylinder such that E = dn. Consider an auxiliary bipartite graph with bipartition X = V and Y = V 1 V 1 and edge set E = {(x, y) X Y : y H(x)}. Then X = n and Y = m = n 1. For x, x X, put d x,x = G(x, x ) /m. Using the induction assumption, H(x, x ) contains n 2( 1) d 2 1 x,x /2 1 copies of K 1 (2). Furthermore, from the previous claim we now that X x x d x,x ( ) dn 2. Then ( ) H = ( X H(x, x ) K (2) x x K 1 (2)) as required. (1 + o(1)) X (1 + o(1)) n2( 1) 2 1 = (1 + o(1)) n2 2 d2, 2 ) (( dn 2 x x n2( 1) 1 ( n 2 The proof of C 3 C 4 is then straightforward. 1 d2 1 x,x ) /( n 2 )) 2 1 Proof of C 3 C 4. The first part (i.e., the inequality) follows from the previous claim. To obtain the asymptotic equality in the case in which the joint lins are almost all (ε, d 2 )-regular, we observe the following. For = 2 we use the fact that (ε, d 2 )-regularity of joint lins means that d x,x d 2 for almost all pairs of vertices x, x V. Then we have the asymptotic equality at every step of equation (46), which is exactly what we need to show. For > 2, since H(x, x ) is (ε, d 2 )-regular for almost all pairs of vertices x, x V, by the induction assumption H(x, x ) contains (1 + o(1))n 2( 1) (d 2 ) 2 1 /2 1 copies of K 1 (2). Hence the number of copies of K (2) containing x, x is (1 + o(1))n 2( 1) (d2 ) 2 1, (46)

24 24 Y. KOHAYAKAWA, V. RÖDL, AND J. SKOKAN and so, summing over all x, x X with x x, we have that the number of copies of K (2) in H is ( ) n (1 + o(1)) n 2( 1) (d2 ) 2 1 = (1 + o(1)) n2 2 d2, as required Proof of C 4 C 3. The proof of this implication will be based on Claims 5.1 and 5.2 and on a standard application of the Cauchy Schwarz inequality. Proof of C 4 C 3. For = 2, this implication follows from the following. Let G = (X Y, E) be a bipartite graph with X = Y = n, and E = dn 2, and assume that G contains (1 + o(1))n 4 d 4 /4 copies of K 2 (2), i.e., ( G K 2(2)) = (1 + o(1))n 4 d 4 /4. But then we must have equality everywhere in (46), which means that d x,x d 2 for almost all pairs of vertices x, x X. This shows, however, that G(x, x ) is (ε, d 2 )-regular for almost all pairs of vertices x, x X. Assume now we have > 2. Let H = (V 1 V, E) be a (, )-cylinder with V i = n for all i [] and E = dn. Suppose that H contains (1 + o(1))n 2 d 2 /2 copies of K (2). Consider an auxiliary bipartite graph with bipartition X = V and Y = V 1 V 1 and edge set E = {(x, y) X Y : y H(x)}. Then X = n and Y = m = n 1. For x, x X, put d x,x = G(x, x ) /m. From Claim 5.1 we obtain ( ) X dn d x x x,x, 2 and so X x x d2 1 ) 2 1 ( ) ( ( ) 1 n n x,x X 2 2 d x x x,x ( ) (( ) /( )) 2 1 n dn n (1 + o(1)) n d2 We apply Claim 5.2 to H(x, x ) and obtain that it contains at least (1 + o(1))n 2( 1) d 2 1 x,x /2 1 copies of K 1 (2). Consequently ( ) H = X {copies of K (2) containing x, x } K (2) x x (1 + o(1)) X n 2( 1) x x 2 (47) (1 + o(1)) n2( 1) n d2 = (1 + o(1)) n2 2 d2. 1 d2 1. (47) x,x (48)

25 HYPERGRAPHS, QUASI-RANDOMNESS, AND REGULARITY 25 On the other hand, by C 4 we have that ( ) H = (1 + o(1)) n2 K (2) 2 d2. (49) From (48) and (49) we conclude that d x,x d 2 for almost all pairs x, x X = V, and therefore that H(x, x ) contains (1 + o(1))n 2( 1) (d 2 ) 2 1 /2 1 copies of K 1 (2). In view of the induction assumption this means that H(x, x ) is (ε, d 2 )- regular for almost all pairs x, x V, i.e., C 3 holds Proof of C 3 C 2. We start with the following claim. Claim 5.3. Let c > 0 be a fixed constant. Let G = (X Y, E) be a bipartite graph with X = n, Y = m, and assume that G(x, x ) c 2 m for almost all pairs x, x X. Then G(x) cm for almost all vertices x X. Proof. Indeed, suppose that G(x) (c+ε)m for all vertices x X, where X X is big. Let G be the subgraph of G induced on X Y and let G (y) = c y X for all y Y. Note that G (x, x ) = G(x, x ) for all x, x X. Then we have X x x G(x, x ) 1 2 c2 m X 2, where we write X x x for the sum over all pairs {x, x } of distinct vertices from X. On the other hand, X G(x, x x x ) = ( G ) (y) 2 y Y y Y 1 2 c2 y X 2. Hence y Y c2 y c 2 m which implies, by the Cauchy Schwarz inequality, that c y c for almost all y Y. But then, cm X c y X = E(G ) = G (x) (c + ε)m X, y Y x X which is a contradiction. The same applies to the set of all vertices x X for which G(x) (c ε)m. Now we give a proof of the implication C 3 C 2. Proof of C 3 C 2. We proceed by induction on. For = 2, the statement follows from Claim 5.3. Let > 2 be given. We shall prove that C 3 C 2 holds for. Thus, assume that the lin H(x, y) is (ε, d 2 )-regular for almost all x, y V. We shall prove that for almost all x V the lin H(x) is (ε, d)-regular, where ε 0 as ε 0. Consider an auxiliary bipartite graph G = (X Y, E) with bipartition X = V and Y = [V 1 ] 2 [V 1 ] 2 and edge set E = {(x, y) X Y : y spans a copy of K 1 (2) in H(x)}. Then X = n and Y = ( n 2) 1 n 2( 1) /2 1. Let x and x be such that H(x, x ) is (ε, d 2 )-regular. Since H(x, x ) is a ( 1, 1)-cylinder, we may apply the implication C 3 C 4 that we have already proved to deduce that H(x, x ) contains n 2( 1) (d 2 ) 2 1 /2 1 = n 2( 1) d 2 /2 1 copies of K 1 (2). This means that almost all pairs of vertices x, x X have their common neighborhood of size