Size Ramsey numbers of triangle-free graphs with bounded degree

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1 Size Ramsey numbers of triangle-free graphs with bounded degree David Conlon Rajko Nenadov Abstract The size Ramsey number ˆr(H) of a graph H is the smallest number of edges in a graph G which is Ramsey with respect to H, that is, such that every two-colouring of the edges of G contains a monochromatic copy of H. A seminal result of Chvátal, Rödl, Szemerédi and Trotter implies that for any there exists a constant b such that ˆr(H) bn 2 for every graph H with n vertices and maximum degree. Recently, Kohayakawa, Rödl, Schacht and Szemerédi obtained the first subquadratic bound for this class of graphs, showing that ˆr(H) = O(n 2 1/ +o(1) ). Our main result is an improvement on this bound, namely, ˆr(H) = O (n 2 1 +o(1)) 1/2, when H is also triangle-free. In particular, this improves the upper bound on the size Ramsey number of bipartite graphs with bounded degree. Following Kohayakawa et al., we work in the random setting, showing that for any there exist constants c and C such that the binomial random graph G(n, p) is with high probability Ramsey for all triangle-free graphs with cn vertices and maximum degree at most, provided p (C log n/n) 1/( 1/2). A crucial point is that this goes below the probability (log n/n) 1/, which is a natural boundary for problems that involve embedding a large graph of maximum degree in G(n, p). Our proof uses the sparse regularity lemma, the recently proven K LR conjecture and ideas of Ferber, Škorić and the authors on universality in random graphs. One of the new ingredients is a deterministic embedding strategy based on an incrementation argument and may be of independent interest. 1 Introduction Given a graph H, a graph G is said to be Ramsey for H if any colouring of the edges of G in red and blue contains a monochromatic copy of H. The classical theorem of Ramsey [34] states that for any graph H there exists N N such that K N is Ramsey for H. The smallest such N, denoted by r(h), is called the Ramsey number. The problem of estimating these numbers has received a great deal of attention over the years and has led to the introduction of many important tools and techniques. However, exact (or even asymptotic) values are known only for a handful of examples. The Ramsey number has a strong dependency on the density of the graph under consideration. Very dense graphs, such as the complete graph K n, are known to have Ramsey number exponential in the number of vertices n. On the other hand, if H is d-degenerate, that is, every induced subgraph of H has a vertex of degree at most d, then a recent breakthrough by Lee [32], solving a conjecture Mathematical Institute, Oxford OX2 6GG, United Kingdom. david.conlon@maths.ox.ac.uk. Research supported by a Royal Society University Research Fellowship and by ERC Starting Grant School of Mathematical Sciences, Monash University, Melbourne, Australia. rajko.nenadov@monash.edu. 1

2 of Burr and Erdős [9], shows that r(h) bn for some constant b = b(d). In other words, such sparse graphs have linear Ramsey number. A weaker form of the Burr Erdős conjecture was established much earlier by Chvátal, Rödl, Szemerédi and Trotter [10]. Using the regularity lemma, they showed that for any positive integer there exists a constant b = b( ) such that if H is a graph with n vertices and maximum degree then r(h) bn. In recent years, the dependency of b on has been greatly improved [13, 11, 19], with many of the ideas developed to tackle this problem contributing directly to Lee s proof (see also [20, 21]). For a more detailed account of these developments and other recent progress in graph Ramsey theory, we refer the reader to the survey by Conlon, Fox and Sudakov [14]. A reader interested in the broader aspects of Ramsey theory may also wish to consult the books [25, 33]. Our concern will be with the notion of size Ramsey number, first introduced by Erdős, Faudree, Rousseau and Schelp [17]. Given a graph H, the size Ramsey number ˆr(H) is the smallest integer m such that there exists a graph G with m edges which is Ramsey for H. The existence of the ordinary Ramsey number immediately implies the upper bound ˆr(H) ( ) r(h) 2. As observed by Chvátal, this is tight when H = K n. To see this, note that a graph G with fewer than ( r(k n) ) 2 edges has chromatic number strictly less than r(k n ). Let V 1,..., V r(kn) 1 be a partition of G into r(k n ) 1 independent sets and consider a two-colouring χ of the edges of K r(kn) 1 without a monochromatic copy of K n. The colouring of G where every edge between V i and V j receives the colour χ(ij) then clearly contains no monochromatic copy of K n. ( When H is the path P n on n vertices, Beck [7] significantly improved the trivial bound ˆr(P n ) r(pn) ) 2, showing that ˆr(Pn ) bn for an absolute constant b. He also raised the question [8] of whether ˆr(H) grows linearly for any sequence of graphs with bounded maximum degree. This was proven for trees by Friedman and Pippenger [22] and for cycles by Haxell, Kohayakawa and Luczak [26]. However, the general case was answered in the negative by Rödl and Szemerédi [36], who showed that there exists a constant c > 0 such that for every sufficiently large n there is a graph H with n vertices and maximum degree 3 for which ˆr(H) n log c n. In the same paper, they conjectured that log c n can be improved to n ε for some constant ε > 0, but this remains open. Suppose now that H is a graph with n vertices and maximum degree at most. By the result of Chvátal et al. discussed above, r(h) O(n), with the implied constant depending only on. Combining this with the estimate ˆr(H) ( ) r(h) 2 shows that ˆr(H) bn 2 for some constant b = b( ). By using random graphs and the sparse regularity method, this simple bound was recently improved by Kohayakawa, Rödl, Schacht and Szemerédi [31], who showed that ˆr(H) bn 2 1/ (log n) 1/ for an appropriate constant b = b( ). Our main result is an improvement to this bound when H is also triangle-free. Note that this includes the case where H is bipartite. Theorem 1.1. For any integer 5, there exists a constant b such that every triangle-free graph H with n vertices and maximum degree at most satisfies ˆr(H) bn 2 1/( 1/2) (log n) 1/( 1/2). The restriction 5 is of a purely technical nature (arising from our Lemma 4.1) and we believe that it can be avoided. However, the requirement that H is triangle-free is more fundamental and removing it will either require delicate technical work or, most likely, new ideas. 2

3 We will derive Theorem 1.1 as a consequence of a stronger theorem, Theorem 1.2 below. One sense in which this theorem is stronger is that it guarantees monochromatic copies of every graph in some large family H rather than just a single graph H. To be more precise, we say that a graph G is H-universal if it contains a copy of every graph H H and H-Ramsey-universal if every two-coloring of the edges of G contains a monochromatic H-universal graph. For now, we will speak principally about the family H (n) of all graphs on at most n vertices with maximum degree at most and its subfamily F (n), consisting of all triangle-free graphs in H (n). Then Theorem 1.2 implies that there exist graphs with bn 2 1/( 1/2) (log n) 1/( 1/2) edges which are F (n)-ramsey-universal. We will find such graphs by showing that for appropriately chosen N and p the binomial random graph G(N, p), where each edge in an N-vertex graph is chosen independently with probability p, has the required property with high probability. By working in this context, we are following Kohayakawa, Rödl, Schacht and Szemerédi [31], who showed that for every integer 2, there exist constants B N and C > 0 such that if N = Bn and p = p(n) (C log N/N) 1/ then lim Pr [G(N, p) is H (n)-ramsey-universal ] = 1. n We prove an analogous result for the subfamily F (n), but with an improved probability. Theorem 1.2. For any integer 5, there exist constants B N and C > 0 such that if N = Bn and p = p(n) (C log N/N) 1 1/2 then lim Pr [G(N, p) is F (n)-ramsey-universal ] = 1. n Since the number of edges in G(N, p) is almost surely at most (1 + ε)n 2 p, Theorem 1.2 implies the existence of a graph G with at most ( ) 1 C log(bn) (1 + ε)n 2 p = (1 + ε)(bn) 2 1/2 Bn = bn 2 1/( 1/2) (log n) 1/( 1/2) edges which is F (n)-ramsey-universal, for a suitable constant b. This proves Theorem 1.1. Perhaps the chief point of interest about Theorem 1.2 is that it remains valid when p is significantly smaller than (log n/n) 1/. This latter probability serves as a natural boundary for problems involving the embedding of large graphs with maximum degree, since it is the lowest probability at which one can expect that every collection of distinct vertices will have many common neighbours, a property that facilitates a greedy embedding strategy. As we will see, going below this probability introduces significant difficulties, only some of which we have been able to resolve. Organization. The paper is organised as follows. In Section 1.1, we give an overview of the proof of Theorem 1.2, highlighting the main ideas and difficulties. Section 2 contains some simple results about random graphs that will be used throughout the paper. Section 3 demonstrates the embedding strategy used in the proof of Theorem 1.2 in a much simpler setting. Specifically, we will show that G(n, p) is almost surely universal for the family of almost-spanning d-degenerate graphs with bounded maximum degree, for a p which is at most a logarithmic factor away from the optimal value. This result might be of independent interest. In Section 4, we show that every graph in F (n) admits a decomposition of a particular type. The precise statement of this decomposition lemma is quite complicated, but it will be very important, when we come to embed the graphs in F (n) into 3

4 G(N, p), that a decomposition of exactly this form exists. Section 5 introduces the sparse regularity method and the main technical machinery behind the proof of Theorem 1.2. Then, in Section 6, we combine the results and methods of the previous sections to complete the proof. Finally, Section 7 mentions some extensions of our result and possible research directions. Notation. Whenever we do not explicitly specify the dependency of a constant on previously defined ones, we implicitly assume it depends on all of them. Moreover, if, for example, a constant a depends on the constants b, c, d in Lemma 5.2, then, throughout the paper, we use a 5.2 (b, c, d) to denote this dependency. Given a graph G, we write V (G) and E(G) for the vertex and edge sets of G, respectively, and let v(g) = V (G) and e(g) = E(G). For a vertex v V (G) and subset Y V (G), we use dist G (v, Y ) to denote the length of the shortest path in G between v and some vertex in Y. For subsets X, Y V (G), let E G (X, Y ) denote the set of edges having one endpoint in X and one in Y, and let e G (X, Y ) = E G (X, Y ) be the number of such edges. Similarly, we use N G (X, Y ) = {y Y : x X such that {x, y} E(G)} to denote the set of all neighbours of X in Y. For brevity, we let N G (X) := N G (X, V (G)). For a vertex v V (G), we write deg G (v, X) = N G ({v}, X) and deg G (v) = N G ({v}) for the number of neighbours of v in X and in the whole graph G, respectively. We use the standard notation (G) and δ(g) for the maximum and minimum degree of vertices in G, respectively. We shall also use the notion of a common neighbourhood, ˆN G (X, Y ) = {y Y : {x, y} E(G) for all x X}. For an integer k N and subsets A, B V (G), we say that A is (k, B)-independent in G if there is no path of length at most k (i.e., a path with at most k + 1 vertices) with endpoints distinct vertices in A and all other vertices in B. Given reals x, y, ε > 0, we write x = (1 ± ε)y if x ((1 ε)y, (1 + ε)y). Similarly, we write x (1 ± ε)y if either x (1 ε)y or x (1 + ε)y. Given a set S, we say that a family of subsets {S 1,..., S t } of S is a partition of S if S i S j = for distinct i, j [t] and S = S 1... S t. Note that we allow sets S i to be empty. We say that an event happens asymptotically almost surely (a.a.s.) or with high probability (w.h.p.) if it happens with probability 1 in the limit (as n ). Finally, to improve readability, we omit floor and ceiling signs whenever they are not crucial. 1.1 Overview of the proof Using the regularity method, Chvátal, Rödl, Szemerédi and Trotter [10] reduced the problem of proving that K N is Ramsey for some H H (n) to proving that certain random-like graphs are universal for H (n) (note that there is no colouring involved in this second problem). With the development of the sparse regularity method, and the results on inheritance of regularity due to Gerke et al. [23] in particular, Kohayakawa, Rödl, Schacht and Szemerédi [31] were able to extend this approach to random graphs. Here we follow the same general strategy, but with a more efficient embedding technique. Consider a graph H H (n) and let G G(N, p), where N = Bn for some constant B > 1. We say that an injective function f : V (H) V (G) is an embedding of H into G if {f(u), f(w)} E(G) 4

5 for every {u, w} E(H). Suppose we wish to find such a function iteratively, vertex by vertex. Let h 1,..., h l be an ordering of the vertices of H and, for each i = 1,..., l, choose f(h i ) as follows: (i) let N i be the set of possible candidates for f(h i ), that is, the vertices in V (G) that are adjacent to the image of each vertex in N H (h i, {h 1,..., h i 1 }), N i := V (G) N G (f(h j )); j<i {h i,h j } E(H) (ii) if F i := N i \ {f(h 1 ),..., f(h i 1 )} is non-empty, set f(h i ) := v i for some v i F i and, otherwise, terminate the procedure with failure. If this procedure does not terminate with failure, the resulting map clearly embeds H into G. Suppose now that H is -regular. Then, regardless of the ordering, we inevitably arrive at the situation where all neighbours of some vertex h i have already been embedded. If p is asymptotically smaller than (log n/n) 1/, the threshold for the property that every subset of vertices has a nonempty common neighbourhood, we cannot ensure that N i, so the procedure can easily terminate in failure. In fact, the situation is worse still, since N i does not guarantee that F i. Because of this, many of the results about embedding large graphs in G(n, p) (see, e.g., [2, 3, 5, 16]) require that p is at least on the order of (log n/n) 1/. Techniques that apply below this probability, such as the second moment method (see, e.g., [35]), are usually less constructive in nature, which can make them difficult to use for universality or resilience-type results. Nevertheless, by combining ideas from these distinct approaches, Conlon, Ferber, Nenadov and Škorić [12] showed that G(n, p) is almost surely H ((1 ε)n)-universal for p n 1/( 1) log 5 n, provided 3. The main idea is to separate every graph H H ((1 ε)n) into two parts: one which is ( 1)-degenerate and another which is composed of cycles of length O(log n). By ordering the vertices of the first part such that each vertex h i has at most 1 neighbours in {h 1,..., h i 1 }, we see, from our earlier discussion, that p (log n/n) 1/( 1) suffices to guarantee that N i. For now, we simply ignore the issue of free space and assume that it is always possible to place f(h i ) in F i whenever N i is non-empty, so that we can embed the first part. To complete the embedding, we use Janson s inequality to embed each of the small cycles as a whole, rather than vertex by vertex. This offers enough leverage that we can push the bound on p significantly below (log n/n) 1/. We apply a similar strategy to prove our embedding lemma, Lemma 6.1. We begin with Lemma 4.1, in which we partition the vertices of H F (n) into two sets V 1 and V 2 such that 1. H[V 1 ] (the subgraph of H induced by the vertices in V 1 ) is almost ( 2)-degenerate and 2. each connected component of H[V 2 ] is either a cycle of length O(log n) or is of constant size and satisfies certain density constraints. The first part is then embedded one vertex at a time and the second part one connected component at a time. We now highlight some of the differences between our approach here and that in [12]. The reason we require ( 2)-degeneracy (instead of 1) is that, in the sparse regularity setting, we will be able to control the edges between two sets only if both are of order Ω(1/p). If there were a vertex h i with N H (h i, {h 1,..., h i 1 }) = 1, we would expect N i np 1, 5

6 which is o(1/p) when p = o(n 1/ ). But this poses a problem if there is an edge {h i, h z } E(H) for some z > i. Indeed, in that case, we would need f(h i ) to have good degree into the set N z = V (G) N G (f(h j )), j<i {h j,h z} E(H) something we cannot guarantee if N i is too small. Unfortunately, we are not able to take H[V 1 ] to be ( 2)-degenerate while keeping the structure of connected components in the second part V 2 under control. What we can show instead is that there is a choice for V 1 and V 2 where the components of H[V 2 ] are well-behaved and there is an ordering of V 1 such that the vertices which have 1 neighbours preceding them in the ordering (instead of at most 2) have a rather special structure. This is spelled out more fully in Lemma 4.1. In order to embed the components of H[V 2 ], we need to use the recently proven K LR conjecture instead of Janson s inequality. This is one of the main reasons why we fail to handle triangles: each of the vertices in a triangle might have 2 previously embedded neighbours, so that their candidate sets are each of order O(np 2 ). However, calculation shows that if we have three sets of order np 2 and p = o(n 1/ ), then the K LR conjecture cannot guarantee that there exists a copy of K 3 touching each of them. Another issue is that the K LR conjecture only gives meaningful information for constant size subgraphs. Fortunately, if the component we wish to embed is not of constant size we know it is a cycle, which allows us to deal with it without resorting to the K LR conjecture. These considerations are summarised in Lemma 5.9. Finally, there is the issue of free space that we have ignored so far. In [12], this was resolved by using a result of Ferber et al. [18] to embed the first part and a hypergraph matching criterion of Aharoni and Haxell [1] to embed the second part. However, we were unable to adapt either of these arguments to our current setting. Instead, we develop a simple greedy strategy for choosing f(h i ) that also allows us to maintain control over the sets F i. The same strategy is then used regardless of whether we embed a single vertex or an entire component. We demonstrate this on a simple example in Section 3. 2 Properties of random graphs In this short section, we prove a couple of simple results about the distribution of edges in random graphs. We will need the well-known Chernoff inequality. Though we state this result for binomial distributions, we note that a similar inequality also holds for hypergeometric distributions [27]. Theorem 2.1 (Chernoff s inequality). Let ε be a positive constant and X B(n, p) a binomial random variable with parameters n and p. Then Pr[ X EX εex] e Ωε(EX). The following proposition will be used in the proof of Theorem

7 Proposition 2.2. For every µ > 0, there exists a constant C > 0 such that if p C/n then G G(n, p) a.a.s. has the property that e G (X, Y ) = (1 ± µ) X Y p for every pair of disjoint sets X, Y V (G) with X, Y µn. Proof. The result follows easily from Chernoff s inequality and taking a union bound over all possible choices of X and Y. We omit the details. The following proposition will be used in the proof of the embedding lemma, Lemma 6.1. Proposition 2.3. For every d N and λ (0, 1), there exist constants C, L > 0 such that if ( ) C log n 1/d p, n then G G(n, p) a.a.s. has the following property: for every family D 2 V (G) of pairwise disjoint d-subsets, { (1 ± λ) D np ˆN(S, d V (G)) =, if D λ/p d (1 ± λ)n, if D L/p d. S D Proof. Consider a (fixed) family D 2 V (G) of pairwise disjoint d-subsets of size D = x λ/p d and let U = V (G) \ S D S. Observe that the bound on p implies that U = n(1 o(1)). For each u U, let X u be the indicator variable for the event that there exists S D with u ˆN(S, U). Since all sets in D are pairwise disjoint, we have Pr[X u = 1] = 1 (1 p d ) x xp d (1 λ/2) and Pr[X u = 1] xp d. Here we used the fact that 1 ab (1 a) b 1 ab + (ab) 2 /2 for every b N and any constant a > 0 such that ab < 1 (this follows from the binomial theorem). Therefore, for X = u U X u, we have EX = nxp d (1 ± 2λ/3). Note that the variables X u are mutually independent and, thus, by applying Chernoff s inequality, we get Pr [ S D ˆN(S, V (G)) (1 ± λ)nxp d ] Pr[X (1 ± 3λ/4)nxp d ] e αnxpd n αcx, where α > 0 depends only on λ. Taking a union bound over all possible choices for D, we obtain the desired upper bound λ/p d n xd n αcx = o(1) x=1 on the probability that there exists a family D which violates the required property, provided C = C(α, d) is sufficiently large. To prove the second part, it suffices to consider families of size x = L/p d. Let D be a family of disjoint d-subsets with D = x and W V (G) a subset of size W = λn. Then W = W \ S D S is of order W = (1 o(1)) W. Thus, Pr[ ˆN(S, W ) = ] Pr[ ˆN(S, W ) = ] = (1 p d ) (1 o(1))λnx e (1 o(1))λnxpd e λnl/2, S D S D 7

8 with room to spare. Taking a union bound over all choices for the subset W and the family D, we see that the probability there exists D and W such that S D ˆN(S, W ) = is at most ( ) n n xd e λnl/2 2 n e log n dl/pd λnl/2 e n+dln/c λnl/2 = o(1), λn for suitably chosen L = L(λ) > 0 and C dl. This implies that S D ˆN(S, V (G)) n λn for every family D of pairwise disjoint d-subsets of size D L/p d, as required. 3 Universality for degenerate graphs In this section, we demonstrate one of the main ideas used in the proof of our embedding lemma, Lemma 6.1. For integers, D, n N, let D (n, D) denote the family of all D-degenerate graphs on at most n vertices and with maximum degree at most. Using Theorem 4.1 in [18], it is easy to show that G(n, p) is almost surely D ((1 ε)n, D)-universal if p n 1/2D+o(1). On the other hand, a simple first moment argument, counting the number of copies of the Dth power of a long path, shows that the probability p must be at least on the order of n 1/D for this property to hold. We will show that this value can be achieved up to a logarithmic factor. Theorem 3.1. For any constant ε > 0 and integers, D N with D, there exists a constant C > 0 such that if ( C log 2 ) 1/D n p, n log log n then G(n, p) is a.a.s. D ((1 ε)n, D)-universal. Proof. Let L and C be constants larger than those given by applying Proposition 2.3 with λ = ε/2 and d taking every possible value in {1,..., D}. By a union bound, we obtain that G G(n, p) a.a.s. satisfies the property given by Proposition 2.3 for all d {1,..., D}. Let 2 log n k = and s = εn/(3k) log log n and consider a random partition V 1,..., V k of V (G) with V i = s for i {2,..., k} and V 1 = n sk (1 ε/2)n. Given a family D of pairwise disjoint d-subsets of V (G) of size at most λ/p d and j {2,..., k}, the expected size of S D ˆNG (S, V j ) is E [ S D ˆN G (S, V j ) ] = S D ˆN G (S, V (G)) s n (1 ε λ) D npd 3k C D log n, where C = C (C, ε, λ) grows with C. Therefore, Chernoff s inequality for hypergeometric distributions and a union bound over all families D and indices j {2,..., k} show that there exists a partition V 1,..., V k with the following property: for every d [D], j [k] and every family D of pairwise disjoint d-subsets, { ε D np d ˆN G (S, V j ) > 6k, if D λ/p d and j {2,..., k} (1) (1 ε)n, if D L/p d and j = 1. S D 8

9 In the first inequality, we have implicitly assumed λ < 1/2 and the second one follows from V 1 (1 ε/2)n. We now describe how this implies that G is D (l, D)-universal, where l = (1 ε)n. Consider an arbitrary graph H D (l, D) and let h 1, h 2,..., h l be an ordering of the vertices of H such that deg H (h i, {h 1,..., h i 1 }) D for every i [l]. Since H is D-degenerate, such an ordering always exists. We iteratively find distinct vertices v 1,..., v l V (G) such that the natural mapping f : V (H) V (G) given by f(h i ) = v i is an embedding. For each i = 1,..., l, choose v i as follows: (i) if S i := N H (h i, {h 1,..., h i 1 }) is the empty set then choose an arbitrary v i V 1 \{v 1,..., v i 1 }; (ii) otherwise, let j i {1,..., k} be the smallest number such that is non-empty and choose an arbitrary v i C i. C i := ˆN G (f(s i ), V j i ) \ {v 1,..., v i 1 } Assuming that the procedure is well-defined (that is, it is indeed possible to make choices as stated), the definitions of S i and C i imply that the resulting sequence of vertices induces an embedding of H into G. We now verify that steps (i) and (ii) are always possible. From V 1 (1 ε/2)n > l, we have that (i) is well-defined. In order to prove that step (ii) is possible, we show (1 ε)n, if j = 1 ( ) V j {v 1,..., v l } g(j), where g(j) = LD 2 /p D, if j = 2 g(j 1)D 2 6k, otherwise, εnp D for every j {1,..., k}. Then, owing to the choice of k, we have g(k) 0, which together with (1) applied to D = {f(s i )} implies that N G (f(s i ), V k )\{v 1,..., v i 1 } for every i [l]. Consequently, j i in (ii) is well-defined. Since v(h) = (1 ε)n, ( ) trivially holds for j = 1. Let us assume, for the sake of contradiction, that there exists j {2,..., k} such that the set of indices J = {i [l] : v i V j } is of order J > g(j). Consider the smallest such j. Since S i D and H has maximum degree at most, it is easy to see (for example, using a greedy procedure) that there exists d {1,..., D} and a subset J J of size { J J D 2 > g(j) L/p d D 2 s, if j = 2 j := g(j 1)6k, if j 3 εnp d such that S i = d and S i S i = for any distinct i, i J. We may assume that J has size exactly s j, observing that if j 3 then s j = o(1/p d ). Therefore, from (1) applied with D = {f(s i )} i J, ˆNG (f(s i ), V j 1 ) > g(j 1). i J 9

10 On the other hand, by the definition of j i in (ii), we have ˆN G (f(s i ), V j 1 ) {v 1,..., v i 1 } for every i J, which implies i J ˆNG (f(s i ), V j 1 ) V j 1 {v 1,..., v l }. However, by the minimality of j, we have V j 1 {v 1,..., v l } g(j 1), which gives a contradiction. This completes the proof of the lemma. 4 Decomposing triangle-free graphs We prove that the vertex set of every graph H F (n) can be partitioned into subsets V and Q such that the induced subgraph H[V ] is almost ( 2)-degenerate and each connected component of H[Q] is either a short cycle or satisfies certain 2-density condition. The 2-density of a graph G = (V, E), denoted by m 2 (G), is defined as { e(g[v } ]) 2 m 2 (G) = max V V V. 1 V 2 This density measure is tightly related to the K LR conjecture, which we introduce in the next section. Properties of the decomposition formally read as follows. Lemma 4.1. Let H F (n) for some 5. Then there exists a partition [ t ] V (H) = ({r i } S i ) Q of the vertices of H such that the following holds for every i [t]: (a) deg H (r i, {r 1,..., r i 1 }) 2, (b) S i N H (r i ), (c) deg H (s, Q) = 1 and deg H (s, {r 1,..., r i }) = 1 for every s S i, and i=1 (d) {r i } S i is (3, t j=i+1 ({r j} S j ) Q)-independent. Moreover, for every connected component Q of H[Q] we have Q is a cycle of size at most 2 log n and/or v(q ) 5 2, δ(q ) 2 and m 2 (Q ) 3/2. Before we give the proof, we briefly discuss properties given by Lemma 4.1. First, the condition 5 is an artifact of the proof and we believe that 3 suffices. Note that the union of {r i } S i s represents V from the preceding discussion. The property (a) implies that H[{r 1,..., r t }] is ( 2)-degenerate. Moreover, in foresight we have that the vertices in V which have 1 embedded neighbours prior to their embedding are exactly vertices in S i s. For them the lemma provides additional structure: the last neighbour of a vertex s S i into {r 1,..., r t } is r i and, by (d), vertices in S i have the strong independence property. By embedding vertices in S i right after 10

11 the vertex r i, this structural information will make it possible to keep the candidate sets under control. Unfortunately, the exact reason why this is the case (and how we do it) is of somewhat technical nature and will become apparent only in the proof of the Embedding Lemma (Lemma 6.1). The minimum degree and the 2-density of connected components in H[Q] spring a relation between the K LR conjecture and the bound on p in Theorem 1.2. In particular, the smaller 2-density of components in H[Q] the better bound on p we get. In order to push p below n 1/, it turns out that anything smaller than 2 suffices. Therefore, one of the reasons why we fail to handle triangles is m 2 (K 3 ) = 2. Proof of Lemma 4.1. Let q N 0 be the largest number for which there exists a family C = {C i } i [q] of disjoint subsets of V (H) such that each H[C i ] is a cycle of size at most 2 log n and e(c i, C j ) = 0 for distinct i, j [q]. If there is more than one such family, choose arbitrary one which minimizes q i=1 C i. Let V C = i [q] C i and set Γ := N H (V C, V (H) \ V C ) and L := V (H) \ (V C Γ). Observe that H[L] is 2-degenerate: otherwise, it contains a subgraph of minimum degree at least 3 and it is easy to show that such subgraph contains an induced cycle of size at most 2 log n. This contradicts the maximality of q. If q = 0, then L = V (H) and by 2-degeneracy there exists an ordering {r 1,..., r v(h) } such that deg H (r i, {r 1,..., r i 1 }) 2 2, thus we are done. Therefore, we may assume q 1 (hence V C ). We now define the set Q. Let {U 1,..., U q } be a family of disjoint subsets of Γ L which maximizes i [q] U i under the following constraints: (a Q ) e(c i U i, C j U j ) = 0 for distinct i, j [q], (b Q ) if C i > 5 then U i =, and (c Q ) if C i 5 then the induced subgraph Q i = H[C i U i ] satisfies m 2 (Q i ) 3/2, δ(q i ) 2 and dist Qi (u, C i ) 2 for every u U i. Since H is triangle-free, each cycle C i is of length at least 4 and one can check that m 2 (C i ) 3/2. Therefore, Q is well defined. Furthermore, note that dist Hi (u, C i ) implies v(q i ) 5 2 and the connected components induced by Q := i [q] (C i U i ) satisfy requirements of the lemma. Let V U := i [q] U i and write Γ 1 Γ\(V U N H (V U )) for the subset of all vertices s with deg H (s, V C ) = 1. Similarly, let L 0 := L \ (V U N H (V U )) and write L 1 L \ V U for the subset of all vertices v with deg H (v, V U ) = 1. We need the following two claims which follow from the maximality of V U : (i) If s 1, s 2 Γ 1 are distinct vertices such that either {s 1, s 2 } E(H) or N H (s 1, L 0 ) N H (s 2, L 0 ), then {s 1, s 2 } is (3, Q)-independent. (ii) If s Γ 1 and v N H (s, L 1 ) then {s, v} is (3, Q)-independent. Before we prove these claims we first finish the proof of the lemma. Let Γ + = Γ \ (Γ 1 V U ) and L + = L \ (V U L 0 L 1 ), and note that deg H (v, Q) 2 for every v Γ + L +. Furthermore, we split L 0 into L + 0 and L 0, with L+ 0 L 0 being the subset of all vertices v L 0 with deg H (s, Γ 1 ) 2 and L 0 := L 0 \ L + 0. Consider a maximal subset Γ 1 Γ 1 such that there exists an ordering {r 1,..., r t } of V = (L \ V U ) Γ + Γ 1 which satisfies (a) and (a R ) vertices in L 0 L 1 precede vertices in L + Γ + L + 0, 11

12 (b R ) deg H (r i, {r 1,..., r i 1 } (L 0 L 1)) 2 for every r i L 0 L 1, and (c R ) vertices in L 0 L 1 L + Γ + precede vertices in L + 0. Note that for Γ 1 = such ordering can easily be obtained from the 2-degeneracy of H[L 0 L 1] and deg H (v, Γ 1 Q) 2 for every v L + 0 L+ Γ +. Therefore, Γ 1 is well defined. Next, observe that S := Γ 1 \ Γ 1 is an independent set in H. Indeed, if there exists an edge {s 1, s 2 } E(H[S]) then setting r t+1 := s 1 contradicts the maximality of Γ 1. The same argument shows deg H(s) = for every s S. We now partition S into S 1,..., S t according to properties (b) and (c) of the lemma: s S i (1 i t) iff i is the largest index such that s N H (r i ). By the definition, each set S i satisfies (b). Since deg H (s, V U ) = 0 for s Γ 1 and S is an independent set, we have deg H (s, V ) = 1. Therefore S 1,..., S t is indeed a partition of S and (c) holds as well. It remains to verify (d). We show a stronger statement: {r i } S i is (3, {r i+1,..., r t } (S \ S i ) Q)-independent for every i [t]. First, observe that for every i [t] with S i we have deg H (r i, {r 1,..., r i 1 }) = 2. (2) Let us assume that this is not true for some r i. Set r j+1 := r j (for t j i) and r i := s for arbitrary s S i. It is easy to see that (a) remains satisfied in the new ordering. Since (a R ) (c R ) are not influenced by this shifting we get a contradiction with the maximality of Γ 1. This immediately gives S i = for r i L + Γ +. Next, consider some r i L 0 and suppose that S i. Then S i = N H (r i, Γ 1 ) (since vertices in L 0 have at most one neighbour in Γ 1 S) and (a R ) imply N H (r i, {r 1,..., r i 1 }) = N H (r i, {r 1,..., r i 1 } (L 0 L 1)). From (b R ) we then conclude deg H (r i, {r 1,..., r i 1 }) 2 3, which contradicts (2). Therefore, for r i L 0 L+ Γ + we have S i = and the property (d) is trivially satisfied. Next, consider r i L 1 (and in parallel r i Γ 1 ) and suppose S i. From deg H (r i, Q) = 1 and (2) we have S i = {s}. Moreover, using additionally the fact that S is an independent set, we also have N H ({r i, s}, {r i+1,..., r t } (S \ {s})) =. (3) (Recall that N H (X, Y ) stands for the union of neighbourhoods of vertices v X in Y, not their common neighbourhood). If {r i, s} is not (3, {r i+1,..., r t } (S \ {s}) Q)-independent then (3) implies {r i, s} is not (3, Q)-independent, which contradicts claim (ii) (claim (i)). Finally, consider r i L + 0 and let us assume S i. Note that (2) implies S i 2. If S i = {s 1, s 2 } then (2) and the fact that S is an independent set further imply N H ({r i, s 1, s 2 }, {r i+1,..., r t } (S \ S i )) =. (4) From the second part of the claim (i) we have that {s 1, s 2 } is (3, Q)-independent, which together with (4) shows that it is also (3, {r i+1,..., r t } (S \ S i ) Q)-independent. Since N H (r i, Q) =, we conclude that r i satisfies (d). Let us now suppose S i = {s}. Without loss of generality we may assume {v} = N H (r i, {r i+1,..., r t } (S \ S i )). Moreover, from N H (r i, Q) = we have that any path of length at most 3 between r i and s with all internal vertices being in {r i+1,..., r t } (S \ {s}) Q induces a path of length at most 2 between v 12

13 and s, with the same restriction on internal vertices. From (c R ) we have v L + 0 Γ 1. If v L+ 0, then the property (d) follows from N H (v, Q) = and N H (s, (S \ {s}) {r i+1,..., r t }) =. Otherwise, if v Γ 1 and the property (d) is violated, from N H(s, (S \ {s}) {r i+1,..., r t } V U ) = we have N H (s, V C ) N H (v, V C ). But this implies {s, v} is not (3, Q)-independent, which contradicts the claim (i). Therefore r i satisfies (d), which finishes the proof of the lemma. It remains to prove claims (i) and (ii). We only give proof of the claim (i), since the claim (ii) follows by the same argument. Proof of the claim (i). By the definition of Γ 1 we have deg H (s j, V C ) = 1 and deg H (s j, V U ) = 0 (j = 1, 2). Therefore, we may assume there is a unique C i C which contains {c 1 } = N H (s 1, V C ) and {c 2 } = N H (s 2, V C ). If {s 1, s 2 } is not (3, Q)-independent, then either c 1 = c 2 or {c 1, c 2 } E(H). In any case, there exists a cycle {s 1, s 2 } C {s 1, s 2 } C i L 0 of size at most 5. Since such cycle has no common edge with any other cycle C j, j i, if C i > 5 then by exchanging C i and C we get a contradiction with the minimality of q i=1 C i. Otherwise, let us consider Q i = H[C i U i {s 1, s 2 } v], where v N H (s 1, L 0 ) N H (s 2, L 0 ) is an arbitrary element if such exists and v = otherwise. It is easy to verify that Q i satisfies (c Q), which contradicts the maximality of V U. Therefore, {s 1, s 2 } is (3, Q)-independent. 5 The regularity method In this section we introduce the machinery used to prove Theorem 1.2. Generally speaking, the Regularity Lemma imposes random-like properties which are useful for finding certain subgraphs. The workflow of the proof of Theorem 1.2 is briefly as follows: We start with an arbitrary colouring of a random graph G G(n, p) and select a subgraph G c induced by a suitable colour c. Since G c might have lost certain properties of the random graph (for example, the structure of edges between large subsets), we use the Regularity Lemma to restore them up to a certain level. Finally, we use these properties to find a copy of the graph under consideration. This approach was pioneered by Chvátal et al. [10]. In the next subsection we give basic definitions and state the version of the Szemerédi s Regularity Lemma suitable for sparse graphs. In Section 5.2 we introduce the relaxed notion of regularity, the so called lower-regularity, and its inheritance with respect to vertex and edge subsets. Finally, in Section 5.3 we state the K LR conjecture and an analogue statement for cycles of logarithmic length. We then use them to derive Lemma 5.9, the main lemma of this section. Later in Section 6 we use Lemma 5.9 to deal with components of H[Q] given by the decomposition result (Lemma 4.1). 5.1 Preliminaries Let G = (V, E) be a graph and p (0, 1]. For subsets X, Y V (not necessarily disjoint), we define the p-density of the pair (X, Y ) as d G,p (X, Y ) = e G(X, Y ) p X Y, 13

14 and write d G (X, Y ) := d G,1 (X, Y ) for the standard definition of the density. If the graph G is clear from the context, we omit it from the subscript. Note that if G G(n, p), then for all sufficiently large subsets X and Y we expect d G,p (X, Y ) = 1 ± o(1). The following definition captures this observation in a more general form. Definition 5.1 ((ε, p)-regularity). Let ε > 0 and 0 < p 1 be given and let G = (V, E) be a graph. For disjoint subsets X, Y V, we say that the pair (X, Y ) is (ε, p)-regular if for all X X and Y Y with X ε X and Y ε Y, we have dg,p (X, Y ) d G,p (X, Y ) ε. If (X, Y ) is (ε, d G (X, Y ))-regular, we simply say that it is (ε)-regular. Moreover, if G is a bipartite graph whose parts form an (ε, p)-regular pair, we say that G is (ε, p)-regular. Note that for p = 1 the definition of (ε, 1)-regularity matches the well-known definition of ε- regularity due to Szemerédi [39]. Before we state a theorem which utilizes the notion of (ε, p)- regularity, we need one more definition: Given a constant ε > 0 and a graph G = (V, E), we say that a partition {V i } t i=0 of V is (ε)-regular (for some t N) if (i) V 0 ε V and V 1 =... = V t, and (ii) all but at most εt 2 pairs (V i, V j ) (with 1 i < j t) are (ε)-regular. The vertex class V 0 is called the exceptional set. It was observed by Kohayakawa and Rödl that Szemerédi s Regularity Lemma [39] can be adapted to the notion of (ε, p)-regularity (see, e.g., [28, 30]). Here we state a simplified version of their result due to Scott [38]. Theorem 5.2 (The sparse Regularity Lemma). Given constants ε > 0 and t 0, l N, there exists a constant T = T (ε, t 0, l) t 0 such that for any graph G = (V, E) and a partition P of V of size P = l, there exists an (ε)-regular partition {V i } t i=0 such that t 0 t T 0 and each V i (for 1 i t) is contained in some set from P. The usual choice for a partition P is to simply put P = {V }. However, if we know that a graph G is for example biparite, then it is convenient to assume that every vertex class of the (ε)-regular partition is a subset of the vertex class of G (we will use this in the proof of Lemma 5.8). For more on the sparse Regularity Lemma, see [24]. 5.2 Lower regularity It turns out that, once we are past the initial application of the sparse Regularity Lemma in the proof of Theorem 1.2, the notion of (ε, p)-regularity is too strong and consequently leads to unnecessary complications. Therefore, we shall mostly work with the following one-sided relaxation of the (ε, p)- regularity. Definition 5.3 ((ε, p)-lower-regularity). Let ε > 0 and 0 < p 1 be given and let G = (V, E) be a graph. For disjoint subsets X, Y V, we say that the pair (X, Y ) is (ε, p)-lower-regular if for all X X and Y Y with X ε X and Y ε Y, 14

15 we have d G,p (X, Y ) 1 ε. If (X, Y ) is (ε, d G (X, Y ))-lower-regular, we say that it is (ε)-lower-regular. If G is a bipartite graph whose parts form an (ε, p)-lower-regular pair, we say that G is (ε, p)-lower-regular. We remark that the usual definition of (ε, p)-lower-regularity requires a lower bound on the p- density of large subsets which is relative to the p-density of the pair. We could have also used this definition, however we believe that the given one will make statements and calculations less cumbersome Vertex subsets The definition of (ε, p)-lower-regularity implies that if (X, Y ) is (ε, p)-lower-regular, then (X, Y ) is (ε/µ, p)-lower-regular for every X X (Y Y ) of size X µ X ( Y µ Y ). The next proposition states this fact for future references. Proposition 5.4. Let (V 1, V 2 ; E) be an (ε, p)-lower-regular bipartite graph, for some ε > 0 and p (0, 1]. Then for any µ > ε, every pair of subsets (X 1, X 2 ) with X i V i and X i µ V i (i = 1, 2) is (ε/µ, p)-lower-regular. Surprisingly, Gerke, Steger, Kohayakawa and Rödl [23] showed that if one replaces every with almost every, then the lower-regularity typically inherits on a much finer scale then given by the definition. Theorem 5.5 (Corollary 3.8 in [23]). Given β, ε (0, 1), there exist constants ε = ε(β, ε ) > 0 and L = L(ε ) > 0 such that for any p (0, 1] the following holds: Let (V 1, V 2 ; E) be an (ε, p)-lower-regular bipartite graph. Then for every q 1, q 2 L/p, all but at most ( )( ) V1 V2 β min{q 1,q 2 } pairs (Q 1, Q 2 ) with Q i V i and Q i = q i (i = 1, 2) are (ε, p)-lower-regular. A one-sided version of Theorem 5.5 suitable for application in random graphs was given in [31]. Here we state a minor modification of that result. It is obtained with the same proof, thus we omit it. Lemma 5.6 (Proposition 15 in [31]). For every integer 3 and constants ε, ζ, γ, α > 0, there exist positive constants ε = ε (, ζ), ε = ε(, ε, ζ) and C, such that if p (C log n/n) 1 1 then G G(n, p) a.a.s satisfies the following property: For every tripartite subgraph Γ = (X, Y, Z; E) G and q αp such that (i) X, Z γnp 2 and Y γnp 3, and q 1 q 2 (ii) (X, Y ) is (ε, q)-lower-regular and (Y, Z) is (ε, q)-lower-regular with respect to Γ, the set of vertices with atypical neighbourhood { Γ(ε, q) := x X deg G (x, Y ) Y q/2 or (N Γ (x, Y ), Z) is not (ε, q)-lower-regular with respect to Γ is of size at most ζ X. } 15

16 5.2.2 Edge subsets Some theorems, like Theorems 5.10 and 5.11 in the next section, require an exact bound on the number of edges in a given (ε)-lower-regular graph. The following lemma comes in handy for applying such results. Lemma 5.7 (Lemma 4.3 in [24]). Given ε > 0, there exists a constant C > 0 such that every (ε)- lower-regular bipartite graph B = (V 1, V 2 ; E) contains a (2ε)-lower-regular subgraph B = (V 1, V 2 ; E ) with e(b ) = m edges, for all m satisfying Cv(B) m e(b). We remark that Lemma 4.3 in [24] is concerned with (ε)-regular graphs, but one easily checks that the same proof goes through in the case of (ε)-lower-regularity. Briefly, the idea is to show that a subset of m edges chosen uniformly at random satisfies lower-regularity with positive probability. Unfortunately, most of the time we shall only work under assumption that a certain bipartite graph B = (V 1, V 2 ; E) is (ε, p)-lower-regular, with no control over its actual p-density. Sampling a random subset of m edges scales down the p-density of a linear subgraph typically by m/e(b). Therefore, if e(b) is bounded away from V 1 V 2 p and there exists a pair (X, Y ) with p-density very close to 1 (we can assume both, as otherwise the graph is (ε )-lower-regular) then the (m/ V 1 V 2 )-density of (X, Y ) in the sampled graph becomes bounded away from 1 (from above) and so we lose the lower-regularity property. We overcome this by showing that every (ε, p)-lower-regular contains a spanning (ε )-lower-regular subgraph with density close to p. Lemma 5.8. Given ε > 0, there exist constants ε > 0 and C > 0 such that for p (0, 1], every (ε, p)-lower-regular graph B = (U 1, U 2 ; E) with U 1 = U 2 = n and np C contains an (ε )-lowerregular subgraph B = (U 1, U 2 ; E ) with e(b ) (1 ε )n 2 p. Proof. Let ξ > 0 be such that 12ξ = (ε ) 3 /2, and consider a partition {V i } i [t] of V (B) obtained by applying the sparse Regularity Lemma with P = {V 1, V 2 } and ε ξ. We prove the lemma for { } 2(1 ξ) ε = min, ξ and C = C 5.7(ξ)t t (1 ξ) 3. Since V 0 ξ 2n and all sets except V 0 are of the same size s := V 1, we have (1 ξ)2n/t s 2n/t. Consequently, the number t j of sets V i U j (j = 1, 2) satisfies (1 2ξ)t/2 t j n/s. For each (ξ)-regular pair (V i, V j ) with V i U 1 and V j U 2, let E ij E B (V i, V j ) be an edge-subset of size E ij = (1 ξ)s 2 p such that (V i, V j ; E ij ) is (2ξ)-lower-regular. This is indeed possible due to Lemma 5.7, since εn s and (ε, p)-lower-regularity of (U 1, U 2 ) imply e B (V i, V j ) (1 ε)s 2 p (1 ξ)s 2 p (1 ξ)(1 ξ) 2 4n2 t 2 p C 5.7(ξ) 2s. The last inequality follows from the choice of C and np C. We now define E as the union of all such E ij. Since the number of (ξ)-regular pairs (V i, V j ) with V i U 1 and V j U 2 is at least t 1 t 2 ξt 2 (1 8ξ)t 2 /4, we obtain E (1 8ξ)t 2 /4 (1 ξ)s 2 p (1 9ξ) t2 4 (1 ξ)2 4n2 t 2 p (1 11ξ)n2 p, as required (the choice of ξ gives 11ξ ε with room to spare). Similar calculation gives an upper bound E n 2 p, thus the density p of B = (U 1, U 2 ; E ) satisfies p p. 16

17 It remains to estimate the number of edges in B between subsets X U 1 and Y U 2 of size X, Y ε n. Since p p, it suffices to show e B (X, Y ) (1 ε ) X Y p. For every V i U 1 and V j U 2 such that (V i, V j ) is (2ξ)-regular (with respect to B ) and X V i, Y V j 2ξs, we have e B (X V i, Y V j ) (1 2ξ) X V i Y V j (1 ξ)p. Ignoring the regularity condition for the moment, summing over all pairs (V i, V j ) for which X V i, Y V j 2ξs gives ( X t 1 2ξs 2ξn)( Y t 2 2ξs 2ξn)(1 3ξ)p ( X 4ξn)( Y 4ξn)(1 3ξ)p X Y (1 3ξ)p 8ξn 2 p. (5) As we might have potentially counted edges between non-regular pairs, to compensate it suffices to subtract ξt 2 s 2 p 4ξn 2 p from (5). Now the choice of ξ gives 12ξn 2 p = ε 2 (ε ) 2 n 2 p ε X Y p, 2 and with room to spare we obtain e B (X, Y ) (1 ε ) X Y p. This verifies that B is (ε )-lowerregular. 5.3 K LR conjecture and cycles Given ( a graph H, integers n, m N and a constant ε > 0, we say that a v(h)-partite graph ) Γ = h V (H) V h, E is (H, n, m, ε)-lower-regular if (i) V h = n for every h V (Γ), and (ii) e(v h, V u ) = m and (V h, V u ) is (ε)-lower-regular for every {h, u} E(H). We are mainly interested in finding an embedding f of H into a (H, n, m, ε)-lower-regular graph Γ such that f(h) V h for every h V (H). We call such an embedding canonical. The following lemma complements the decomposition result from Section 4 (Lemma 4.1) by providing the existence of subgraphs with certain density/structural properties. We use m 2 ( ) as defined in Section 4. Lemma 5.9. Given integers D 1 and 2 and reals α, γ > 0, there exist constants ε = ε(d,, α), C > 0 such that if p (C log n/n) 1 1/2, then G G(n, p) a.a.s has the following property: for every graph H with m 2 (H) 3/2 and v(h) D or H is a cycle of size at most 2 log n, if Γ G is an (H, s, m, ε)-lower-regular subgraph with s = γnp 2 / log n and m = αs 2 p, then there exists a canonical embedding of H into Γ. 17

18 The proof of Lemma 5.9 follows easily from Theorem 5.10 and Lemma 5.11, which we state next. The following theorem was first conjectured by Kohayakawa, Luczak and Rödl [29]. It has attracted a lot of attention until it was finally resolved, independently, by Saxton and Thomason [37] and Balogh, Morris and Samotij [6] (see also [15] for a slightly different statement). Theorem 5.10 (K LR conjecture). Given a graph H and a constant β > 0, there exist constants ε 0, C > 0 and n 0 N such that for n n 0, m Cn 2 1/m 2(H) and ε (0, ε 0 ], for all but at most ( ) n β m 2 e(h) m (H, n, m, ε)-lower-regular graphs Γ there exists a canonical embedding of H into Γ. We remark that the theorem proven in [6, 37] deals with the case where the pairs of Γ corresponding to edges of H are (ε)-regular, rather than (ε)-lower-regular. However, careful analysis of their proofs reveals that only the number of edges m and lower-regular properties are used, thus it implies the version stated here. One of the drawbacks of Theorem 5.10 is that it can only be applied to fixed graphs. To deal with cycles of logarithmic size in Lemma 5.9 we need the following statement about expansion in lower-regular graphs, due to Gerke et al. [23]. To state it concisely, we use the following notion of expansion: given l N and δ > 0, we say that a graph G = (V, E) is (δ, l)-expanding for subsets A, B V if for at least (1 δ) A vertices a A there exist at least (1 δ) B vertices in B which are at distance at most l from a. In the next lemma, we use P l to denote the path on l vertices. Lemma 5.11 (Lemma 5.9 in [23]). Given integer l 2 and constants β, δ > 0, there exist constants ε 0, C > 0 and n 0 N such that for all n n 0, m Cn 2 (l 2)/(l 1) and ε (0, ε 0 ], all but at most ( ) n β m 2 l 1 m (P l, n, m, ε)-lower-regular graphs Γ = (V 1,..., V l ; E) are (δ, l 1)-expanding for V 1 and V l. With Theorem 5.10 and Lemma 5.11 at hand, we are ready to prove Lemma 5.9. Proof of Lemma 5.9. Let β = (2α/e 2 ) D2 and assume that ε (0, 1/2) is smaller than any ε 0 (H, β) given by Theorem 5.10 with v(h) D and ε 0 (l = 4, β, δ = 2/3) given by Lemma We first show that G G(n, p) a.a.s does not contain any of the excluded lower-regular configurations from Theorem Let H be a graph on at most D vertices and m 2 (H) 3/2. A simple calculation shows that for sufficiently large enough C > 0 we have p C 5.10 (H, β)s 2/3 /α and consequently m αs 2 p C 5.10 (H, β)s 2 m 2(H). Therefore, the expected number of copies of (H, s, m, ε)-lower-regular graphs excluded by Theorem 5.10 is at most ( ) n v(h) ( ) s β m 2 e(h) ( es p e(h)m e v(h)s log n β m 2 ) e(h)m p ( < (eβ) m e ) e(h)m = o(1). s m m α In the last inequality we used a simple estimate e(h) D 2. Markov s inequality implies that a.a.s none of these copies appear in G. Moreover, since there are only constantly many graphs on at most 18