REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS

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1 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS PETER ALLEN*, JULIA BÖTTCHER*, JOZEF SKOKAN*, AND MAYA STEIN Abstract. Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox and Zhao [Adv. Math , 06 90]. These inheritance lemmas also imply improved H-counting lemmas for subgraphs of bijumbled graphs, for some H.. Introduction Over the past 40 years, the Regularity Method has developed into a powerful tool in discrete mathematics, with applications in combinatorial geometry, additive number theory and theoretical computer science see [4, 7, 0, 3] for surveys. The Regularity Method relies on Szemerédi s celebrated Regularity Lemma [7] and a corresponding Counting Lemma. Roughly speaking, the Regularity Lemma states that each graph can almost be partitioned into a bounded number of regular pairs. More precisely, a pair U, W of disjoint sets of vertices in a graph G is ε- regular if, for all U U and W W with U ε U and W ε W, we have du, W du, W ε, where du, W := eu, W / U W is the density of the pair U, W and eu, W is the number of edges between U and W in G. The Regularity Lemma then says that every graph G has a vertex partition V... V m into almost equal-sized sets such that all but at most εm pairs V i, V j are ε-regular and m is bounded by a function depending on ε but not on G. The Counting Lemma complements the Regularity Lemma and states that in systems of regular pairs the number of copies of any fixed graph H is roughly as predicted by the densities of the regular pairs. In particular, if H is a graph with V H = [m] := {,..., m} and G is an m-partite graph with partition V... V m of V G such that V i, V j is ε-regular whenever ij EH, then the number of labelled copies of H in G with vertex i in V i for each i V H is dvi, V j ± γ i [m] V i, as long as ε is sufficiently small. ij EH Such a Counting Lemma can easily be proved with the help of the fact that neighbourhoods in dense regular pairs are large and therefore inherit regularity. More precisely, if X, Y, Y, Z and X, Z are ε-regular and have density d ε then for most vertices x X it is true that Nx Y = d±ε Y and Nx Z = d±ε Z. Hence one can easily deduct from ε-regularity that the pair Nx Y, Z is ε -regular this is called one-sided inheritance and the pair Nx Y, Nx Z is ε -regular this is called two-sided inheritance for some ε. Using this regularity Date: June 9, 06. * Department of Mathematics, London School of Economics, Houghton Street, London WCA AE, U.K. {p.d.allen j.boettcher j.skokan}@lse.ac.uk. Department of Mathematics, University of Illinois at Urbana-Champaign, 409 W. Green Street, Urbana, IL 680, USA. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile mstein@dim.uchile.cl. Supported by Fondecyt Regular and Millenium Nucleus Information and Coordination in Networks.

2 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 inheritance, the Counting Lemma follows by induction on the number of vertices m of H. For sparse graphs G, that is, G with n vertices and on edges, the error term in the definition of ε-regularity is too coarse, and hence the Regularity Method is, as such, not useful for such graphs. There are, however, sparse analogues of the Regularity Lemma, which rescale the error term and hence are meaningful for sparse graphs. Definition sparse regularity. Let p > 0 and G be a graph. Let U, W V G be disjoint. The p-density of U, W is d p U, W := eu, W /p U W. The pair U, W is ε, p-regular if, for all U U and W W with U ε U and W ε W, we have d p U, W d p U, W ε. It is ε, d, p-regular if, moreover, d p U, W d ε. An ε-regular pair U, W is an ε, p-regular pair with density du, W = p. The Sparse Regularity Lemma see [5, 5] states that any graph can be partitioned into ε-regular pairs and very few irregular pairs. However, a corresponding Counting Lemma for ε-regular pairs is not true in general: One can construct, say, balanced 4-partite graphs such that every pair of parts induces an ε, d, p-regular pair with ε d, but which do not contain a single copy of K 4 see, e.g., [9, p.]. Nevertheless, Counting Lemmas are known for sparse graphs G with additional structural properties. In the case that G is a subgraph of a random graph establishing such a Counting Lemma was a famous open problem, the so-called K LR- Conjecture [6], which was settled only recently [6, 8, 4]. Proving an analogous result for subgraphs G of pseudorandom graphs has been another central problem in the area. The study of pseudorandom graphs was initiated by Thomason [8, 9] see also [] for more background information on pseudorandom graphs, who considered a notion of pseudorandomness very closely related to that of bijumbledness. Definition bijumbled. A pair U, V of disjoint sets of vertices in a graph Γ is called p, γ-bijumbled in Γ if, for all pairs U, V with U U and V V, we have eu, V p U V γ U V. A graph Γ is said to be p, γ-bijumbled if all pairs of disjoint sets of vertices in Γ are p, γ-bijumbled in Γ. A bipartite graph Γ with partition classes U and V is p, γ-bijumbled if the pair U, V is p, γ-bijumbled in Γ. After partial results were obtained in [8], Conlon, Fox and Zhao [9] recently proved a general Counting Lemma for subgraphs of bijumbled graphs. This Counting Lemma has various interesting applications for subgraphs of bijumbled graphs, including a Removal Lemma, Turán-type results and Ramsey-type results. For obtaining Counting Lemmas for sparse graphs the most straightforward approach is to try to mimic the strategy for the proof of the dense Counting Lemma outlined above. The main obstacle here is that in sparse graphs it is no longer true that neighbourhoods of vertices in regular pairs are typically large and therefore trivially induce regular pairs they are of size pn εn. One can overcome this difficulty by establishing that, under certain conditions, typically these sparse neighbourhoods nevertheless inherit sparse regularity. Inheritance Lemmas of this type were first considered by Gerke, Kohayakawa, Rödl, and Steger [3]. Conlon, Fox and Zhao [9] proved Inheritance Lemmas for subgraphs of bijumbled graphs. The main results of the present paper are Inheritance Lemmas which require weaker bijumbledness conditions. The first result establishes one-sided regularity inheritance.

3 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 3 Lemma 3 One-sided Inheritance Lemma. For each ε, d > 0 there are ε, c > 0 such that for all 0 < p < the following holds. Let G Γ be graphs and X, Y, Z be disjoint vertex sets in V Γ. Assume that X, Y is p, cp 3/ X Y -bijumbled in Γ, Y, Z is p, cp log p / Y Z -bijumbled in Γ, and Y, Z is ε, d, p-regular in G. Then, for all but at most at most ε X vertices x of X, the pair N Γ x Y, Z is ε, d, p-regular in G. Comparing this result with the analogue by Conlon, Fox, Zhao in [9, Proposition 5.], we need Γ to be a factor p log p / less jumbled when X = Y = Z. The second result establishes two-sided regularity inheritance under somewhat stronger bijumbledness conditions. Lemma 4 Two-sided Inheritance Lemma. For each ε, d > 0 there are ε, c > 0 such that for all 0 < p < the following holds. Let G Γ be graphs and X, Y, Z be disjoint vertex sets in V Γ. Assume that X, Y is p, cp X Y -bijumbled in Γ, X, Z is p, cp 3 X Z -bijumbled in Γ, Y, Z is p, cp 5/ log p Y Z -bijumbled in Γ, and Y, Z is ε, d, p-regular in G. Then, for all but at most ε X vertices x of X, the pair N Γ x Y, N Γ x Z is ε, d, p-regular in G. Here Γ needs to be a factor p less jumbled when X = Y = Z than in [9, Proposition.3]. We remark that the bijumbledness conditions in our results imply that these implicitly are statements about sufficiently large graphs see Lemma 7... Applications. Counting Lemmas. The most obvious application of our results here is to prove stronger Counting Lemmas than those in [9]. Recall that for a dense graph G and fixed H the Counting Lemma provides matching upper and lower bounds on the number of copies of H in G. By contrast, when G is a subgraph of a sparse bijumbled graph Γ, we formulate two separate Counting Lemmas. The one-sided Counting Lemma gives only a lower bound on the number of copies of H in G, while the twosided Counting Lemma gives in addition a matching upper bound. The motivation for formulating two separate lemmas is that for many graphs H, the bijumbledness requirement on Γ to prove a one-sided Counting Lemma is significantly less than to prove a two-sided Counting Lemma, and for many applications the one-sided Counting Lemma suffices. The statements and proofs of our Counting Lemmas are quite technical, and we prefer to leave them as an Appendix to this paper. Comparison with the results of [9] is unfortunately also not straightforward, in part because the two-sided Counting Lemma in [9] actually provides better performance than the one-sided Counting Lemma there in some important cases, such as for cliques. Briefly, our one-sided Counting Lemma always performs at least as well as either of [9, Theorems. and.4], and in some cases our results are better. For example, if H consists of 0 copies of K 3 sharing a single vertex, then our one-sided Counting Lemma requires p, cp 3 -jumbledness to lower bound the number of copies of H, whereas the results Somewhat confusingly, the terms one-sided/two-sided refer to completely different aspects in the one-sided/two-sided Counting Lemmas and the one-sided/two-sided Inheritance Lemmas. Both are standard terminology.

4 4 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 in [9] require p, cp 4 -bijumbledness. Our two-sided Counting Lemma sometimes performs better than [9, Theorem.]. Again, for 0 copies of K 3 sharing a vertex, we require p, cp 0.5 -bijumbledness while [9] requires p, cp -bijumbledness. In general, our results perform better when there are vertices of exceptionally high degree. For many interesting graphs such as d-regular graphs for any d 3 the performance is identical. Of course, these counting lemmas can also be immediately applied in the relatively straightforward applications presented in [9]. For most of these applications what one requires is a one-sided Counting Lemma. In particular, by using the onesided Counting Lemma resulting from our Inheritance Lemmas the bijumbledness requirements for the removal lemma [9, Theorem.], the Turán result [9, Theorem.4], and the Ramsey result [9, Theorem.6] can always be matched, and in some cases be improved. Blow-up Lemmas and their applications. In addition, our Regularity Inheritance Lemmas also have other applications, in which they perform better than the results from [9] would. Blow-up Lemmas are another important tool in the Regularity Method, which make it possible to derive results about large or even spanning subgraphs in certain graph classes see, e.g., []. In [3] a Blow-up Lemma which works relative to sparse jumbled graphs is proved. The proof of this lemma relies on the Regularity Inheritance Lemmas, Lemmas 3 and 4. As an application of this Blow-up Lemma for jumbled graphs in [] resilience problems for jumbled graphs with respect to certain spanning subgraphs are considered. Such resilience problems see, e.g., [6]; resilience was investigated also earlier, for example as fault-tolerance in [5] received much interest recently. In [] Lemmas 3 and 4 together with the Blow-up Lemma for jumbled graphs are used to derive the following sparse version of the Bandwidth Theorem proved for dense graphs in [7]. Theorem 5. [] For each ε > 0,, and k, there exists a constant c > 0 such that the following holds for any p > 0. Given γ cp max4,3 +/ n, suppose Γ is a p, γ -bijumbled graph, G is a spanning subgraph of Γ with δg k k +ε pn, and H is a k-colourable graph on n vertices with H and bandwidth at most cn. Suppose further that there are at least c p 6 γ n vertices in V H that are not contained in any triangles of H. Then G contains a copy of H. Note that the bijumbledness requirement implicitly places a lower bound on p. It is necessary to insist on some vertices of H not being in any triangles of H, but the number c p 6 γ n comes from the requirements of Lemma 4, and improvement there would immediately improve this statement. This is a very general resilience result, covering for example Hamilton cycles, clique factors, and much more. Note that although a Hamilton cycle might not be -colourable, in [] a more complicated variant of the above statement is proved which allows occasional vertices to receive a k + st colour. Resilience theorems in jumbled graphs. The Andrásfai-Erdős-Sós Theorem states that any n-vertex triangle-free graph with minimum degree greater than 5n is bipartite. Thomassen [30] further proved that the chromatic threshold of the triangle is 3, or in other words that for every ε > 0, any n-vertex triangle-free graph with minimum degree 3 +ε n has chromatic number bounded independently of n. The analogous statements relative to random graphs are false, and there can be a few bad edges destroying bipartiteness, or bounded chromatic number, respectively. In [4] accurate estimates are proved for the number of these bad edges that can exist. On the other hand, one can ask what minimum degree does force triangle-free subgraphs of random graphs, or more generally H-free subgraphs of

5 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 5 random graphs, to have bounded chromatic number; this question is studied in []. In both papers, a key tool is the Regularity Inheritance Lemma for random graphs though in both cases more ideas are needed. It would be interesting to use the results of this paper to give jumbled graph versions of the results of [, 4]... Optimality. Our one-sided Inheritance Lemma is probably not optimal. The optimality of our two-sided Inheritance Lemma would follow from a conjecture of Conlon, Fox and Zhao [9], since an improvement on the bijumbledness requirement in this lemma would imply an improved version of the triangle removal lemma in bijumbled graphs. However, we think it unlikely that this lemma is optimal and believe there is room for improvement in our proof strategy. In the case when H is a clique, Conlon, Fox and Zhao [9] are able to obtain a onesided counting lemma with a bijumbledness requirement matching ours by using a completely different strategy. In particular, when H is a triangle, these counting lemmas imply a triangle removal lemma for subgraphs of bijumbled graphs with β = op 3 n. Such a result was obtained earlier already in [9], where it was also conjectured that this can be improved to β = op n. Conlon, Fox and Zhao [9] conjecture the contrary. We sympathise with the former conjecture, and believe that it would be extremely interesting to resolve this question. Organisation. The remaining sections of this paper are devoted to the proofs of the Inheritance Lemmas. We start in Section with an overview of these proofs. Section 3 collects necessary auxiliary results on bijumbled graphs and sparse regular pairs. In Sections 4 and 5 we prove various lemmas used in the proofs of the Inheritance Lemmas: Section 4 establishes lemmas on counting copies of C 4 in various bipartite graphs, and Section 5 concerns a classification of pairs of vertices in such graphs according to their codegrees. In Section 6 we prove Lemma 3 and in Section 7 Lemma 4. Notation. For a graph G = V, E we also write V G for the vertex set and EG for the edge set of G. We write eg for the number of edges of G. For vertices v, v V and a set U V we write N G v; U and N G v, v ; U for the G-neighbourhood of v in U and common G-neighbourhood of v and v in U, respectively. Similarly, deg G v; U := N G v; U and deg G v, v ; U := N G v, v ; U. If U = V we may omit U and, if G is clear from the context, we may also omit G. For disjoint vertex sets U, W V the graph G[U, W ] is the bipartite subgraph of G containing exactly all edges of G with one end in U and the other in W. We write eu, W for the number of edges in G[U, W ].. Proof Overview We sketch the proof of Lemma 3 first. We label the pairs in Y as typical, heavy, or bad, according to whether their G-common neighbourhood in Z is not significantly larger than one would expect, or so large as to be unexpected even in Γ, or intermediate. By using the bijumbledness of Y, Z in Γ we can show that the heavy pairs are so few that one can ignore them Lemma 6. Now suppose that x X is such that N Γ x; Y, Z is either too dense or is not sufficiently regular. In either case, by several applications of the defect Cauchy- Schwarz inequality, we conclude that N Γ x; Y, Z contains noticeably more copies of C 4 in G than one would expect if Y, Z were a random bipartite graph of the same density Lemma 3. In particular, the average pair of vertices in N Γ x; Y has noticeably more G-common neighbours in Z than one would expect. It follows

6 6 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 that a substantial fraction of the pairs y, y in N Γ x; Y are bad or heavy. Since there are few heavy pairs, we see that there are many bad pairs Lemma 7. On the other hand, because Y, Z is regular, we can count copies of C 4 in G crossing the pair Lemma 3. A further application of the defect Cauchy-Schwarz inequality tells us that a very small fraction of the pairs in Y are bad, and using the bijumbledness of X, Y we conclude that there are few triples x, y, y such that xy and xy are edges of Γ and y, y is bad Lemma 8. Putting these two statements together, we conclude that there are few x X such that N Γ x; Y, Z is either too dense or is not sufficiently regular. By averaging, if there are few dense pairs there are also few pairs which are too sparse. This completes the proof of Lemma 3. The proof of Lemma 4 is very similar. We have to additionally classify the pairs in Y as typical, heavy or bad with respect to x X, which we do according to their G-common neighbourhood in N Γ x; Z. Now Lemma 7 as before tells us that if x X is such that N Γ x; Y, Z is either too dense or is not sufficiently regular, then a substantial fraction of the pairs y, y in N Γ x; Y are bad with respect to x. Lemma 8 continues to tell us that there are few triples x, y, y such that xy and xy are edges of Γ and y, y is bad, and Lemma 6 continues to tell us that we can ignore the heavy pairs. To complete the argument as before, it remains to show that if y, y is a typical pair, then there are few x such that xy, xy Γ and y, y is bad with respect to x. To prove this we do not use the requirement xy, xy Γ, but simply bound, using bijumbledness of X, Z, the number of x with abnormally many neighbours in N G y, y ; Z. This step is where we require most bijumbledness. We believe it is wasteful, but were not able to find a more efficient way. 3. Preliminaries 3.. Bijumbledness. One consequence of a pair U, V being p, γ-bijumbled is that most vertices in U have about p V neighbours in V. Lemma 6. Let k, c > 0, and 0 < p <, and let U, V be a p, c p k U V - bijumbled pair in a graph Γ. Then, for any γ > 0, we have {u U : deg Γ u; V ± γp V } c p k γ U. Proof. Let U + := {u U : deg Γ u; V > + γp V }. By bijumbledness applied to the pair U +, V we have + γp U + V < eu +, V p U + V + c p k U V U + V. Simplifying this gives U + c p k γ U. A similar calculation for the set U of vertices in U with fewer than γp V neighbours in V yields the same bound on U, and the result follows. Moreover, non-trivial bijumbled graphs cannot be very small. Lemma 7. Let 0 < c 4, 0 < p 4 and k. Let Γ be a graph, and let U, V be p, c p k U V -bijumbled in Γ. Then we have U, V 8 c p k. Proof. By Lemma 6, the number of vertices in U with more than p V neighbours in V is at most c p k U U. It follows that we can take a set U U of min { 4 p, U } vertices, each with degree at most p V. The union of their neighbourhoods covers by definition at most 4 p p V = V vertices of V, so

7 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 7 we can let V be a subset of V vertices in V with no edges between U and V. Applying bijumbledness to the pair U, V, we have 0 = e U, V p U V c p k U V U V, which implies c p k U V p U V = p min { 4 p, U } V. Hence, we obtain U c p k min { 4 p, U }. The inequality U 4 c p k U is false for all U by our choice of c, p and k, so we conclude that The same bound applies to V. U 8 c p k. Remark 8. Erdős and Spencer [0] see also Theorem 5 in [] observed that there exists c > 0 such that every m-vertex graph with density p contains two disjoint sets X and Y for which ex, Y p X Y c pm X Y, as long as p p /m. One can also recover Lemma 7 using this result. See also Remark 6 in [8]. 3.. Sparse regularity. The Slicing Lemma, Lemma 9, states that large subpairs of regular pairs remain regular. Lemma 9 Slicing Lemma. For any 0 < ε < γ and any p > 0, any ε, p-regular pair U, W in G, and any U U and W W with U γ U and W γ W, the pair U, W is ε/γ, p-regular in G with p-density du, W ± ε. In the other direction, the following lemma shows that, under certain conditions, adding a few vertices to either side of a regular pair cannot destroy regularity completely. Lemma 0. Let 0 < ε < 0 and c 0 ε3. Let G be a spanning subgraph of a graph Γ, let U, V be a pair of disjoint sets in V Γ, and let U U and V V. Assume U, V is p, cp U V -bijumbled in Γ and U, V is ε, d, p- regular in G. If U + 0 ε3 U and V + 0 ε3 V, then U, V is ε, d, p-regular in G. Proof. Let X U with X ε U and Y V with Y ε V be arbitrary. Using p, cp U V -bijumbledness of U, V in Γ we have ex U, Y \ V e Γ X U, Y \ V p U ε3 0 V + cp U V U ε3 0 V 5 ε3 p U V. Similarly, we have ex \ U, Y p ε3 0 U + ε3 0 V + cp U V ε 3 0 U + 0 ε3 V 5 ε3 p U V. Moreover, since U, V is ε, d, p-regular, ex U, Y V = d ± εp X U Y V. Hence ex, Y = ex U, Y V + ex U, Y \ V + ex \ U, Y = d ± εp X U Y V ± 5 ε3 p U V = d ± 3 ε p X U Y V = d ± εp X Y. We conclude that U, V is ε, d, p-regular in G.

8 8 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, Cauchy-Schwarz. We use the following defect form of the Cauchy-Schwarz inequality. This inequality and a proof can be found in [, Fact B]. Lemma Defect form of Cauchy-Schwarz. Let a,..., a k be real numbers with average at least a. If for some δ 0 at least µk of them average at least + δa, then k a i ka + µ µδ. i= and the same bound is obtained if at least µk of the a i average at most δa. 4. Counting copies of C 4 in regular, irregular and dense pairs The following counting lemma for counting C 4 in ε, d, p-regular subgraphs of bijumbled graphs is as given by Conlon, Fox, and Zhao [9, Proposition 4.3]. We write C 4 G for the number of unlabelled copies of C 4 in G. Lemma counting C 4 in regular pairs. For any ε > 0, c > 0 and d [0, ] the following holds. If U, V is a p, cp U V -bijumbled pair in a graph Γ, and G is a bipartite subgraph of Γ with parts U and V which forms an ε, d, p-regular pair, then C 4 G = 4 d 4 ± 00c + ε / p 4 U V. The next lemma gives a lower bound on the number of copies of C 4 in a bipartite graph of a given density. Moreover, if this bipartite graph is not ε-regular we obtain an even stronger lower bound. Observe that for this lemma we do not require that the pair is a subgraph of a pseudorandom graph. Lemma 3 counting C 4 in dense pairs and irregular pairs. Let 0 < ε l3 0 3, let G be a bipartite graph with vertex classes U and V of sizes m n ε 9 l3 respectively. Suppose that G has density q ε 0 l3 n /. a We have C 4 G ε 8 l3q 4 4 m n. b If G is not ε l3 -regular, then we have C 4 G + ε 3 l3q 4 4 m n. Proof. Assume G has density q. Clearly, we have degu, u C 4 G =. {u,u } U Hence, for bounding this quantity we will analyse common neighbourhoods of vertices in U. Let us first bound the average a := m u u U degu, u. Observe that if a +ε 8 l3q n then, using Jensen s inequality and facts that q ε 4 l3n / and m n ε 9 l3, we get m + ε 8 C 4 G l3 q n + ε8 l3 + ε 9 l3q 4 n m + ε 8 l3q 4 4 n m, and thus are done. Hence we may assume in the following that a + ε 8 l3q n. For obtaining a corresponding lower bound on a note that the average degree of the vertices in V is qm. Hence by Jensen s inequality we have degv qm qmqm ε0 n = n n l3q m, v V

9 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 9 where the second inequality uses q q ε 0 l3 m. Therefore degu, u = degv n {u,u } U v V This gives 3 a ε 0 l3q n. Moreover, we obtain from and that 4 C 4 G {u,u } U ε0 l3q m m ε 0 l3q degu, u nq m. n. For estimating the sum of squares in this inequality we will use the defect form of Cauchy-Schwarz Lemma. Let us first establish the first part of Lemma 3. Lemma with k = m, µ = δ = 0 so actually without defect implies that {u,u } U Hence, by 4 we have C 4 G degu, u m ε 0 ε 0 l3q 4 n ε 8 l3q 4 4 m n, m a 3 m l3 q 4 n nq m m n m ε 0 l3 q 4 n. q 4 ε 0 l3 n / as desired, where we used q ε 0 l3 n / in the second inequality. For the second part of the lemma, we will use a similar calculation, but we will apply Lemma with µ, δ > 0. So we need to find a subset Ũ U of vertices whose average pair degrees differ significantly from a. The following definition will be useful. For a set Ũ U, let 5 aũ := Ũ degu, u = {u,u } Ũ Ũ v V degv, Ũ. Claim 4. If G is not ε l3 -regular, then there is a set Ũ U with Ũ ε l3m such that aũ + ε5 l3q n + ε 5 l3a, where the second inequality follows from. Before we prove this claim, let us show how it implies the second part of our lemma. For this, assume that G is not ε l3 -regular, and let Ũ be the set guaranteed by Claim 4. Since Ũ ε l3m, there are at least ε l3 m ε m l3 / pairs of vertices in Ũ. Thus we can use Lemma with k := m, µ = ε l3 / and δ = ε 5 l3

10 0 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 to infer that {u,u } U degu, u m Together with 4 this gives the desired C 4 G m + ε m + ε a + ε l3 3 m ε 0 l3 q 4 n + ε l3 m + ε l3 q 4 n. 4 l3 4 l3 5 m q 4 n nq q 4 n + ε 3 l3q 4 4 n m, where again we used q ε 0 l3 n / in the second inequality. It remains to prove the claim. Proof of Claim 4. Since G is not ε l3 -regular there are sets U U and V V with U = ε l3 m and V ε l3 n such that either 6 du, V > + ε l3 q or du, V < ε l3 q. Now we distinguish three cases. First suppose that du, V + ε3 l3 0 q =: q. Then using again Jensen s inequality we have au εl3 m degv, U = v V ε l3m n qε l3m qε l3m ε l3m n ε7 l3 q ε l3m = ε 7 l3 q n + ε3 l3 5 q n, where the second inequality uses q q ε 8 l3/m and the last inequality uses ε l Hence we can choose U as Ũ. Secondly, suppose that du, V ε3 l3 0 q and let U := U \ U. Then du, V = du, V nm du, V ε l3 nm ε l3 nm q ε 3 l3 0 qε l3 ε l3 + ε4 l3 q. 0 Using an analogous calculation as in the previous case we obtain au + ε 5 l3q n and thus can choose U as Ũ. Finally, suppose ε3 l3 0 q < du, V < + ε3 l3 0 q. In this case we will use Ũ := U and apply Lemma to bound 7 au ε l3m degv, U degv, U. v V v V

11 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 For this observe that 8 b := degv, U = n n du, V ε l3 mn = v V On the other hand bv := ε l3 n thus, by 6, we obtain v V ± ε3 l3 0 qε l3 m. v V degv, U = ε l3 n du, V ε l3mn and bv > + ε l3 qε l3 m + ε l3 b or bv < ε l3 qε l3 m ε l3 b. Therefore Lemma applied with k := n, δ := ε l3 /, µ := ε l3, and with b instead of a implies that degv, U n ε3 l3 q ε 0 l3m ε 3 l3 + + ε3 l3 q n ε 4 ε l3 00 l3m. Together with 7 and 8 this gives au + ε3 l3 q n 00 + ε3 l3 0 as desired, where we used q 400ε 4 l3/m. qn ε l3 m + ε3 l3 q n Typical pairs, bad pairs, heavy pairs The proofs of our inheritance lemmas rely on estimating the number of copies of C 4 which use certain types of vertex pairs in one part of a regular pair which is a subgraph of a bijumbled graph. We will consider vertex pairs that are atypical for the regular pair, which we call bad, and vertex pairs which are even atypical for the underlying bijumbled graph, which we call heavy. Definition 5 bad, heavy pairs. Let G be a graph and U and V be disjoint vertex sets in G. Let q [0, ] and δ > 0. We say that a pair uu of distinct vertices in U is V, q, δ-bad in G if Moreover, uu is V, q-heavy in G if deg G u, u ; V + δq V. deg G u, u ; V 4q V. Pairs which are neither heavy nor bad with certain parameters will usually be called typical. In a bijumbled graph we can establish good bounds on the number of copies of C 4 which use heavy pairs. Lemma 6 C 4 -copies using heavy pairs. Let Γ be a bipartite graph with partition classes U and V that is p, c p 3/ log p / U V -bijumbled. Assume further that for all u U we have deg Γ u; V p V. Then the number of copies of C 4 in Γ which use a pair in U which is V, p-heavy in Γ is less than 64c p 4 U V. Proof. We first fix u U and count the number of copies of C 4 in Γ which use a pair that contains u and is V, p-heavy. Let W u U \ {u} be the set of vertices u U \ {u} such that uu is a V, p-heavy pair. We now split W u according to the number of common neighbours the vertices of W u have with u. Since 4p V degu, u p V for all u W u, we can partition W u into W u = S... S log p with S t = { u W u : t 4p V Nu, u < t 4p V }

12 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 for t [ log p ]. Since deg Γu p V, we can take a superset N u V of Nu of size p V. Applying Lemma 6 to U, N u with c replaced by c log p /, k = and γ = t we see that the number of vertices in U with at least + γp N u = + γ p V neighbours in N u is at most c log p p t U. Since each vertex of S t has at least 4 t p V > + γ p V neighbours in Nu N u, we conclude that S t 3 t c p U log p 3 t c U log p. For a fixed u, the number of copies of C 4 using u and u is Nu,u Nu, u. Hence, the total number of copies of C 4 using u and any vertex of S t is at most S t t 4p V 64c p 4 U V log p. Summing over the at most log p values of t, we conclude that the total number of copies of C 4 in Γ using u and some u W u is at most 64c p 4 U V. Finally, summing over all u U, the total number of copies of C 4 in Γ using V, p-heavy pairs in U is at most 64c p 4 U V as desired. Using this lemma we obtain a good lower bound on the number of bad pairs in subgraphs of bijumbled graphs which are irregular or exceed a certain density. Lemma 7 many bad pairs. Given d 0, and ε 0 3, if δ ε 9 /0, ε ε 9 d/00 and c d ε 0 /00 then for any p 0, the following holds. Let Γ be a p, c p 3/ log p / U V -bijumbled graph and let G be a bipartite subgraph of Γ with vertex classes U and V. Assume further that deg Γ u; V p V for all u U. If i U, V has density at least d εp and is not ε, p-regular in G, or ii U, V has density at least d + ε p in G, then at least ε 0 d 4 U pairs uu U are V, dp, δ-bad in G. Proof. Let P h be the set of V, p-heavy pairs in Γ and P b be the set of V, dp, δ-bad pairs in G which are not in P h. Let P t := U \Pb P h. Denote by C4 h the number of those copies of C 4 in G that use a pair in P h, and define C4 b and C4 t similarly. We claim that, if i or ii are satisfied, then 9 C 4 G + ε 9 d 4 p 4 4 U V. Indeed, Lemma 7 implies that U and V are of size at least 8 c log p / p 8 c p, and hence ν 0 := min{ U, V } / 8c p. Now assume first that i holds. Then G has density at least d εp ε 0 8c p ε 0 ν 0 because c d ε 0 /00 and ε ε. But this is the condition we require to apply Lemma 3 with ε l3 = ε and q = d εp to G. By Lemma 3b we have C 4 G + ε 8 d ε 4 p 4 4 U V + ε 9 d 4 p 4 4 U V, where we used ε ε 9 d/00 in the second inequality. Hence 9 holds in this case. If, on the other hand, ii holds, then G has density at least d + ε p dp ε 0 8c p ε 0 ν 0 where we used c d ε 0 /00 in the second inequality. Hence Lemma 3a with ε l3 = ε and q = d + ε p gives C 4 G ε 8 d + ε 4 p 4 4 U V + ε 9 d 4 p 4 4 U V,

13 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 3 because ε ε 9 d/00, which means we also get 9 in this case. Our next goal is to obtain a lower bound for C4. b For this purpose observe that since deg Γ u; V p V for all u U Lemma 6 applies and we obtain C4 h 64c p 4 U V. Moreover, each pair uu P t lies in at most deg G u,u ;V +δd p V + δ d 4 p 4 V copies of C 4 in G by definition of P t. Hence U C4 t + δ d 4 p 4 V 4 + 3δd4 p 4 U V. Thus we conclude from 9 that C4 b = C 4 G C4 h C4 t ε 9 d 4 64c 3δd4 p 4 U V 4 4 8ε 0 d 4 p 4 U V, where we use ε 0 3, c d ε 0 /00, δ ε 9 /0, and ε ε. Now observe that each pair uu in P b is in at most degγ u, u ; V 4p V < 8p 4 V copies of C 4 in G by definition of P b. It follows that as desired. P b 8ε 0 d 4 p 4 U V 8p 4 V = ε 0 d 4 U, The next lemma provides an upper bound for the number of bad pairs in neighbourhoods. Lemma 8 few bad pairs. Let d, δ > 0, let c ε 0 0 δ 6 d 8, and p 0,. Let G Γ and let U, V, W V Γ be disjoint sets such that i U, V is p, c p 3/ U V -bijumbled in Γ, ii V, W is p, c p V W -bijumbled in Γ, and ε, d, p-regular in G, iii and each v V has deg Γ v; U = ± εp U. Then for the sets P b u of pairs vv N Γ u;v which are W, dp, δ-bad in G we have u U P b u δp U V. Proof. Let P b be the set of all pairs vv V which are W, dp, δ-bad in G. Our first step is to obtain an upper bound on P b. Claim 9. P b δ V. Proof of Claim 9. We conclude from Lemma 7 applied to V, W that V, W 8 c p 3 8 p ε. This implies 0 V ε V and d εp V εd εp V, which we will use to estimate binomial coefficients. Let µ be such that P b = µ V. Our goal is to get an upper bound on µ. For this purpose we shall use the defect form of Cauchy-Schwarz, Lemma, to get a lower bound on the number of C 4 -copies in V, W in terms of µ, and combine this with the upper bound on the number of C 4 -copies in regular pairs provided by Lemma.

14 4 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 For the application of Lemma set a vv and define V a := a vv = vv V := deg G v, v ; W for each vv V, V w W degg w; V to be the average of the a vv. Let us now first establish some bounds on a. Since V, W is ε, d, p-regular in G, all but at most ε W vertices of W have at least d εp V neighbours in V. This gives a V d εp V ε W 0 ε d ε p W =: a. On the other hand, by Lemma 6 the number of vertices w W with deg Γ w; V > p V is at most c p W. We conclude that V p V V a W + c p W 4 + c p W 5p W. Now we apply Lemma with k = V and a, δ and µ as given. By the avv average at least a. Moreover, by definition all µk pairs vv P b are W, dp, δ-bad in G, that is, a vv + δd p W + δa by. Lemma thus guarantees that a vv ka + µδ µ vv V V ε 4 d ε 4 p 4 W + µδ 0 ε5 d ε 4 + µδ p 4 V W d 3ε4 + µδ p 4 V W, since ε 5 d ε 4 d 3ε 4. Hence the number of C 4 -copies in V, W in G is avv = a vv vv V vv V vv V a vv 4 d 3ε4 + µδ p 4 V W V 5 p W 4 d 4ε4 + µδ p 4 V W, where we used W 8 p ε in the last inequality. On the other hand, since V, W is ε, d, p-regular in G, and p, c p V W -bijumbled in Γ, Lemma implies that the number of C 4 -copies in V, W in G is at most 4 d c + ε / p 4 V W. Putting these two inequalities together we obtain d 4ε 4 + µδ d c + ε /, and because c ε 0 0 δ 6 d 8 by assumption, we get µ δ as desired. Now for each v V, let V v := {v V : vv P b }. Note that vv P b u if and only if vv P b and u N Γ v; U and uv EΓ. It follows that P b u = e Γ Vv, N Γ v; U. u U v V

15 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 5 Since U, V is p, c p 3/ U V -bijumbled in Γ, we have for each v V that e Γ Vv, N Γ v; U p V v deg Γ v; U + c p 3/ U V V v deg Γ v; U + εp V v U + c p 3/ U V + εp U V = + ε V v + c + ε V p U, where we use assumption iii for the second inequality. We therefore obtain P b u + ε Vv + c + ε V p U u U v V + εp P b U + c p V U V + εδp U + c p V U δp U V, as desired, where in the third inequality we use Claim One-sided inheritance To prove Lemma 3 we combine Lemma 7 and Lemma 8. The former asserts that any vertex x such that N Γ x; Y, Z is not ε, d, p-regular in G creates many pairs in N Γ x, Y which are bad in Y, Z G, whereas the latter upper bounds the sum over x X of the number of such bad pairs. Proof of Lemma 3. We may assume without loss of generality that 0 < ε < 0 4. Given in addition d > 0 set 3 δ = 0 4 ε d 4, ε = 0 6 ε d 6 δ 6 and c = 0 4 ε 0 d 4 δ. As a preparation we first clean up the partition classes X, Y, Z as follows. We let Y Y be the set of vertices y of Y with 4 deg Γ y; X = ± εp X, deg Γ y; Z = ± εp Z, and deg G y; Z = d ± εp Z. Observe that by Lemma 6 and by ε, d, p-regularity of X, Y in G we have 5 Y \ Y c pε Y + c p ε Y + ε Y 3 3ε Y. Hence, X, Y is p, 3 cp3/ X Y -bijumbled in Γ. We then let X X be the set of vertices x of X with 6 deg Γ x; Y = ± εp Y and deg Γ x; Y \ Y 4εp Y. Similarly as before, we apply Lemma 6 once to X, Y with γ = ε and once to the pair X, Y \ Y in X, Y with γ = 3 and use 3 and 6 to obtain 7 X \ X 3 c pε X + 3c p X εp X. By Lemma 9 and because of 4 and 7 it follows that 8 Y, Z is ε, d, p-regular in G and deg Γ y; X = ± 3εp X for each y Y. Moreover, Y, Z is p, 3 cp log p / Y Z -bijumbled in Γ. Thus for each x X, because Y deg Γ x; Y /p ε by 6, 9 Y, Z is p, cp 3/ log p / N Γ x; Y Z -bijumbled in Γ. Finally, let X be the set of vertices in X such that N Γ x; Y, Z is not ε, d, p - regular in G.

16 6 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 We claim that N Γ x; Y, Z is ε, d, p-regular in G for all x X \ X. In order to show this we apply Lemma 0 with ε l0 = ε and c l0 = c, and with U = N Γ x; Y, U = N Γ x, Y and V = V = Z. This is possible by 9, the definition of X, and because ε 3 3εp Y U \ U deg Γ x; Y \ Y 6 4εp Y ε 3 degγ x; Y ε 3 U, 0 0 where for the second to last inequality we use 5 and 6. We conclude that indeed N Γ x; Y, Z is ε, d, p-regular in G. Therefore, by 7 it suffices to show that X ε X to complete the proof. For this purpose, we define for each x X { P b x := yy NΓ x; Y : yy is Z, dp, δ-bad in G and determine a lower bound on P b x in terms of X with the help of Lemma 7 and an upper bound in terms of X with the help of Lemma 8. For the lower bound, fix x X. By 4 the density of N Γ x; Y, Z in G is at least d εp. Hence, by 3, 4, 9 and the definition of X we may apply Lemma 7 with parameters d, ε = ε, δ, ε l7 = ε, c = c and p to the pair NΓ x; Y, Z in G, in the bijumbled graph Y, Z in Γ, using condition i of this lemma. We obtain P b x ε 0d 4 deg Γ x; Y, and therefore x X P b x x X P b x 6 X }. ε 0d 4 ε p Y. For the upper bound we use Lemma 8 with input δ, ε l8 = 3ε, and c = c, and setting U = X, V = Y and W = Z, which we may do by 3, 5, 7 and 8. The conclusion is that x X P bx δp X Y. Together with this gives ε 0d 4 ε X δ X δ X and therefore by 3 we indeed have X ε X. 7. Two-sided inheritance The proof of Lemma 4 follows a similar pattern to that of Lemma 3. Proof of Lemma 4. Assume without loss of generality that 0 < ε < 0 4. Given d > 0, we set ε = 0 0 ε 9 d, δ = 0 0 d 4 ε 3, ε = 0 0 ε 0 δ 6 d 8 and c = 0 3 d ε 0 δ. We now clean up the partition classes X, Y, Z as follows. First, let Y Y be the set of vertices y Y with 3 deg Γ y; Z = ± εp Z, and deg Γ y; X = ± εp X. By Lemma 6 and we have 4 Y \ Y c log p p 3 ε Y + c p ε Y ε Y. We let X X be the set of vertices x X with 5 deg Γ x; Y = ± εp Y, deg Γ x; Y \ Y εp Y, and deg Γ x; Z = ± εp Z. Again, by Lemma 6 and we have 6 X \ X 8c p ε X + c p 4 ε X εp X.

17 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 7 By Lemma 9, by 3 and by 6, we obtain 7 Y, Z is ε, d, p-regular in G and deg Γ y; X = ± 3εp X for each y Y. Moreover, Y, Z is p, cp 5/ log p / Y Z -bijumbled in Γ. Thus for each x X, because Y deg Γ x; Y /p ε and Z deg Γ x; Z/p ε by 5, 8 Y, Z is p, cp 3/ log p / N Γ x; Y N Γ x; Z -bijumbled in Γ. For x X let Define Y x := N Γ x; Y and Z x := N Γ x; Z. X := { x X : d G Y x, Z x d ε p and Y x, Z x G is not ε, d, p -regular }, X := { x X : d G Y x, Z x d + ε p }, and let X := X X. Finally, let X be the set of x X \ X such that Yx, Z x has density less than d ε p in G. We claim that N Γ x; Y, Z x is ε, d, p-regular in G for all x X \X X. This again follows from Lemma 0, which we apply with ε l0 = ε and c l0 = c, and with U = Y x, U = N Γ x, Y, V = V = Z x. This is possible by 8, because Yx, Z x is ε, d, p-regular in G by the definition of X and X, and because U + 0 ε 3 U by a calculation analogous to 0. We conclude that indeed N Γ x; Y, Z x is ε, d, p-regular in G. Therefore, by 6 it suffices to show that X 3 ε X and X 3 ε X to complete the proof. We start with the former. For each x X, let { Pb x := yy Yx : yy is Z x, dp, δ } -bad in G. To bound X, we will again estimate x X P b x in two different ways. The first part is given by the following claim. Claim 0. x X P b x x X P b x 0 0 ε 0 d 4 p X Y. Proof. This bound will follow from Lemma 7. We first need to clean up the pairs Y x, Z x for the application of this lemma. Let Y x Y x consist of the vertices y Y x with deg Γ y; Z x p Z x. The pair Y x, Z x is p, cp 3/ / log p Yx Z x - bijumbled since Y x Z x = ± ε p Y Z by 5. So Lemma 6 and imply Y x \ Y x 8c p Yx log p εp Y x. Moreover, 9 Y x, Z x is p, 4cp 3/ log p / Y x Z x -bijumbled. We now first consider vertices x X. Lemma 7 to Y x, Z x, using condition i of Lemma 7. For this purpose we will first show that Y x, Z x is also not ε 4, d, p -regular in G. Indeed, by and 9 To bound Pb x we want to apply we can apply the contrapositive of Lemma 0 with ε l0 = ε 4 and c l0 = 4c, and with U = Y x, U = Y x, V = V = Z x because Y x \ Y x εp Y x εp Y x 0 ε /4 3 Y x. Since Y x, Z x is not ε, d, p -regular in G, this lemma implies that Y x, Z x is also not ε 4, d, p -regular in G as claimed. Hence, by, 9, the definition of X and the definition of Y x we may apply Lemma 7 to Y x, Z x with input d, ε l7 = ε 4, δ, ε l7 = ε, c = 4c and p. We conclude that P b x ε 4 0d 4 Y x 0 0 ε 0 d 4 p Y.

18 8 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 It remains to consider x X. In this case we want to use Lemma 7ii. To obtain the required density condition, observe that e Γ Y x \ Y x, Z x p εp Y x Z x + cp 3/ log p / Yx Z x εp Y x Z x 3 ε p Y x Z x. Since d G Y x, Z x d+ε p, it follows that d G Y x, Z x d+ 3 ε p by. Hence, by and 9 we can apply Lemma 7 to Y x, Z x with input d, ε l7 = 3 ε, δ ε l7 = ε and c = 4c and conclude that P b x ε 3 0 d 4 Y x 0 0 ε 0 d 4 p Y. Summing over all x X = X X, and using the claim follows. The next claim establishes a complementing upper bound for x X P b x. Claim. x X P b x δp X Y. Proof. In order to estimate x X P b x we will distinguish between the contribution made to this sum by the pairs { Y P b := yy : yy is } Z, dp, δ -bad in G and that made by the pairs P t := Y \ Pb. For the former let P b x := {yy Y x : yy is Z, dp, δ -bad in G}. We use the very rough bound Pb x P b P b x, x X x X which holds since Pb x P b P b x for all x X. By Lemma 8 applied to X, Y, Z with parameters d, δ l8 = δ, c = c ε l8 = 3ε, which we can do by, 7 and since Y, Z is ε, d, p-regular in G, we thus have 30 Pb x P b P b x δp X Y δp X Y. x X x X For the contribution of P t on the other hand, define V b yy := {x X : deg Γ x, NG y, y ; Z + δd p Z x } for yy Y. Observe that we have yy Pb x for some x X if and only if x N Γ y, y ; X and x V b yy. It follows that Pb x P t V b yy. x X yy P t Now let yy P t be fixed. We have deg G y, y ; Z + δ d p Z by definition of P t. Let Z yy be a superset of N G y, y ; Z of size + δ d p Z. By assumption, X, Z is p, cp 3 X Z -bijumbled, and so X, Z yy is p, cd p X Z yy - bijumbled. Lemma 6, with parameters γ = ε, c = cd, k =, then gives 3 {x X : deg Γ x; Z yy + εp Z yy } c d p ε X. Since + δd p Z x 5 + δd p εp Z + εp Z yy and by the choice of Z yy, the left-hand side of 3 is at least V b yy. Summing over all yy P t we conclude x X P b x P t yy P t V b yy c d p ε X Y δp X Y.

19 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 9 Together with 30 this proves the claim. Claims 0 and imply 0 0 ε 0 d 4 p X Y δp X Y and hence 3 X δd 4 ε 30 X ε X 3 ε X, by. It remains to bound X. Let X := X \ X. Claim. X ε X 3 ε X. Proof. Let µ := X X. We bound µ by considering the number T of triples xyz with x X, y Y, z Z which are such that xy, xz EΓ and yz EG. Observe that T = x X e G Yx, Z x and e G Y x, Z x = d G Y x, Z x deg Γ x; Y deg Γ x; Z 5 d G Yx, Z x +ε p Y Z for each x X. Since X X = we get by the definition of X and X T X d ε p + X \ X d + ε p + ε p Y Z 33 = µd ε + µd + ε + ε p 3 X Y Z d µε + ε + 3ε p 3 X Y Z. For obtaining a lower bound on T, let Y Y be the set of vertices y Y with deg G y; Z d εp Z and deg Γ y; X εp X. Since Y, Z is ε, d, p-regular in G and X, Y is p, cp X Y -bijumbled in Γ, applying Lemma 6 we obtain Y \ Y ε Y + 8c p ε Y ε Y by. Now, each y Y contributes at least T y := e Γ NΓ y, X, N G y, Z triples to T. As X, Z is p, cp 3 X Z -bijumbled the definition of Y thus implies that T y p εp X d εp Z cp 3 X Z εp X d εp Z d 3εp 3 X Z, for each y Y. We conclude that T T y Y ε Y d 3εp 3 X Z d 0εp 3 X Y Z. y Y Together with 33 this gives d µε + ε + 3ε d 0ε and so µ ε + 3εε ε by as desired. Claim and 3 prove the lemma. References [] P. Allen, J. Böttcher, J. Ehrenmüller, and A. Taraz, The bandwidth theorem for sparse random graphs, manuscript in preparation. [] P. Allen, J. Böttcher, S. Griffiths, Y. Kohayakawa, and R. Morris, Chromatic thresholds in dense random graphs, submitted. [3] P. Allen, J. Böttcher, H. Hàn, Y. Kohayakawa, and Y. Person, Blow-up lemmas for sparse graphs, In preparation. [4] P. Allen, J. Böttcher, Y. Kohayakawa, and B. Roberts, Triangle-free subgraphs of random graphs, submitted. [5] N. Alon, M. Capalbo, Y. Kohayakawa, V. Rödl, A. Ruciński, and E. Szemerédi, Universality and tolerance extended abstract, 4st Annual Symposium on Foundations of Computer Science Redondo Beach, CA, 000, IEEE Comput. Soc. Press, Los Alamitos, CA, 000, pp. 4. [6] J. Balogh, R. Morris, and W. Samotij, Independent sets in hypergraphs, J. Amer. Math. Soc. 8 05, no. 3, [7] J. Böttcher, M. Schacht, and A. Taraz, Proof of the bandwidth conjecture of Bollobás and Komlós, Math. Ann ,

20 0 P. ALLEN, J. BÖTTCHER, J. SKOKAN, AND M. STEIN JUNE 9, 06 [8] D. Conlon, W. T. Gowers, W. Samotij, and M. Schacht, On the K LR conjecture in random graphs, Israel J. Math , no., [9] D. Conlon, J. Fox, and Y. Zhao, Extremal results in sparse pseudorandom graphs, Adv. Math , [0] P. Erdős and J. Spencer, Imbalances in k-colorations, Networks 97/7, [] P. Erdős, M. Goldberg, J. Pach, and J. Spencer, Cutting a graph into two dissimilar halves, J. Graph Theory 988, no., 3. [] P. Frankl and V. Rödl, Extremal problems on set systems, Random Structures Algorithms 0 00, no., [3] S. Gerke, Y. Kohayakawa, V. Rödl, and A. Steger, Small subsets inherit sparse ϵ-regularity, J. Combin. Theory Ser. B , no., [4] S. Gerke and A. Steger, The sparse regularity lemma and its applications, Surveys in combinatorics 005, London Math. Soc. Lecture Note Ser., vol. 37, Cambridge Univ. Press, Cambridge, 005, pp [5] Y. Kohayakawa, Szemerédi s regularity lemma for sparse graphs, Foundations of computational mathematics Rio de Janeiro, 997, Springer, Berlin, 997, pp [6] Y. Kohayakawa, T. Luczak, and V. Rödl, On K 4 -free subgraphs of random graphs, Combinatorica 7 997, no., [7] Y. Kohayakawa and V. Rödl, Szemerédi s regularity lemma and quasi-randomness, Recent Advances in Algorithms and Combinatorics, CMS Books in Mathematics / Ouvrages de mathématiques de la SMC, Springer New York, 003, pp [8] Y. Kohayakawa, V. Rödl, M. Schacht, P. Sissokho, and J. Skokan, Turán s theorem for pseudo-random graphs, J. Combin. Theory Ser. A 4 007, no. 4, [9] Y. Kohayakawa, V. Rödl, M. Schacht, and J. Skokan, On the triangle removal lemma for subgraphs of sparse pseudorandom graphs, An irregular mind, Bolyai Soc. Math. Stud., vol., János Bolyai Math. Soc., Budapest, 00, pp [0] J. Komlós, A. Shokoufandeh, M. Simonovits, and E. Szemerédi, The regularity lemma and its applications in graph theory, Theoretical aspects of computer science Tehran, 000, Lecture Notes in Comput. Sci., vol. 9, Springer, Berlin, 00, pp. 84. [] M. Krivelevich and B. Sudakov, Pseudo-random graphs, More sets, graphs and numbers, Bolyai Soc. Math. Stud., vol. 5, Springer, Berlin, 006, pp [] D. Kühn and D. Osthus, Embedding large subgraphs into dense graphs, Surveys in combinatorics 009, London Math. Soc. Lecture Note Ser., vol. 365, Cambridge Univ. Press, Cambridge, 009, pp [3] V. Rödl and M. Schacht, Regularity lemmas for graphs, Fete of combinatorics and computer science, Bolyai Soc. Math. Stud., vol. 0, János Bolyai Math. Soc., Budapest, 00, pp [4] D. Saxton and A. Thomason, Hypergraph containers, Invent. Math. 0 05, no. 3, [5] A. Scott, Szemerédi s regularity lemma for matrices and sparse graphs, Combin. Probab. Comput. 0 0, no. 3, [6] B. Sudakov and V. H. Vu, Local resilience of graphs, Random Structures Algorithms , no. 4, [7] E. Szemerédi, Regular partitions of graphs, Problèmes combinatoires et théorie des graphes Colloq. Internat. CNRS, Univ. Orsay, Orsay, 976, Colloq. Internat. CNRS, vol. 60, CNRS, Paris, 978, pp [8] A. Thomason, Pseudorandom graphs, Random graphs 85 Poznań, 985, North-Holland Math. Stud., vol. 44, North-Holland, Amsterdam, 987, pp [9], Random graphs, strongly regular graphs and pseudorandom graphs, Surveys in combinatorics 987 New Cross, 987, London Math. Soc. Lecture Note Ser., vol. 3, Cambridge Univ. Press, Cambridge, 987, pp [30] C. Thomassen, On the chromatic number of triangle-free graphs of large minimum degree, Combinatorica 00, no. 4, Appendix A. Counting Lemmas In this appendix we formulate a sparse one-sided Counting Lemma and a sparse two-sided Counting Lemma requiring stronger bijumbledness, which both follow from our Inheritance Lemmas. Given a graph H with V H = [m], a graph G, and vertex subsets V,..., V m of V G, we write nh; G for the number of labelled copies of H in G with i in V i for each i. Observe that the quantity nh; G depends on the choice of the sets

21 REGULARITY INHERITANCE IN PSEUDORANDOM GRAPHS JUNE 9, 06 V,..., V m, but this choice will always be clear from the context. Given 0 < p, we write dh; G := d p V i, V j. ij EH Again, this quantity depends on the choice of V,..., V m, and again this will always be clear from the context. Still for any given graph H with vertex set [m], which we think of as having order,..., m, and given u, v [m], we define N + v := { w N H v: w > v }, N v := { w N H v: w < v }, N <u v := { w N H v: w < u }. Finally, we let k reg H be the smallest number with the following properties for each i m. For each j i such that ij EH 3 if k > i: jk EH if k > i: jk, ik EH k reg H N i + N <i j + 3 if k > i: jk, ik EH and N <i k N <i j otherwise and for each j, j i such that ij, jj EH k reg H N <i j + N <i j + {.50 if ij EH,.00 otherwise. Informally, the idea is that p, cp k regh -bijumbledness is enough to use Lemmas 3 and 4 to find copies of H in G one vertex at a time, in the natural order,..., m. The following lemma formalises this. Lemma 3 One-sided Counting Lemma. For every graph H with V H = [m] and every γ > 0, there exist ε, c > 0 such that the following holds. Let G and Γ be graphs with G Γ, and let V,..., V m be subsets of V G. Suppose that for each edge ij H, the sets V i and V j are disjoint, and the pair V i, V j is ε, p-regular in G and p, cp kregh V i V j -bijumbled in Γ. Then we have nh; G dh; G γ p eh V i. i V H The proof of this lemma is similar to the proof of [9, Lemma X]. It is also contained in the proof of Lemma 4 below, so we omit the details. The jumbledness requirement in our two-sided Counting Lemma depends on another graph parameter, which is also different from the parameter in the twosided Counting Lemma in [9] and may appear somewhat exotic at first sight. We shall later compare this parameter to other more common graph parameters. Let H be given with vertex set [m], which again we think of as having the order,..., m. For each v V H, let τ v be any ordering of N + v such that N <v w is decreasing. We define dh := max v V H N v + max τv w + N <v w. w N + v The idea is that this parameter controls the bijumbledness we require in order to prove an upper bound on the number copies of H in Γ. In order to count in G, we need both to be able to do this and to use our inheritance lemmas, and we need to consider the same order on V H for both.