A generalized Turán problem in random graphs

Transcription

1 A generalized Turán problem in random graphs Wojciech Samotij Clara Shikhelman June 18, 2018 Abstract We study the following generalization of the Turán problem in sparse random graphs Given graphs T and H, let ex Gn, p, T, H be the random variable that counts the largest number of copies of T in a subgraph of Gn, p that does not contain H We study the threshold phenomena arising in the evolution of the typical value of the random variable, for every H and an arbitrary 2-balanced T Our results in the case when m 2 H > m 2 T are a natural generalization of the Erdős Stone theorem for Gn, p, which was proved several years ago by Conlon and Gowers and by Schacht; the case T = K m has been recently resolved by Alon, Kostochka, and Shikhelman More interestingly, the case when m 2 H m 2 T exhibits a more complex and subtle behavior Namely, the locations of the possibly multiple thresholds are determined by densities of various coverings of H with copies of T and the typical values of ex Gn, p, T, H are given by solutions to deterministic hypergraph Turán-type problems that we are unable to solve in full generality Our arguments involve a somewhat nonstandard application of the hypergraph container method 1 Introduction The well-known Turán function is defined as follows For a fixed graph H and an integer n, we let exn, H be the maximum number of edges in an H-free 1 subgraph of K n This function has been studied extensively and generalizations of it were offered in different settings see [32] for a survey Erdős and Stone [11] determined exn, H for any nonbipartite graph H up to lower order terms Theorem 11 [11] For every fixed nonempty graph H, 1 exn, H = 1 χh 1 + o1 n 2 Note that if H is bipartite, then the theorem only tells us that exn, H = on 2 In fact, the classical result of Kővári, Sós, and Turán [25] implies that in this case exn, H = On 2 c for some c > 0 that depends only on H Two natural generalizations of Theorem 11 have been considered in the literature First, instead of maximizing the number of edges in an H-free subgraph of the complete graph with n vertices, one can consider only H-free subgraphs of some other n-vertex graph G One natural choice is to let G be the random graph Gn, p, that is, the random graph on n vertices whose each pair of vertices 1 A graph is H-free if it does not contain H as a not necessarily induced subgraph 1

2 forms an edge independently with probability p This leads to the study of the random variable ex Gn, p, H, the maximum number of edges in a H-free subgraph of Gn, p Considering the intersection between the largest H-free subgraph of K n and the random graph Gn, p, one can show that if p exn, H 1, then whp ex Gn, p, H 1 + o1 exn, Hp 1 The above bound is not always best-possible If p decays sufficiently fast so that the expected number of copies of some subgraph of H that contain a given edge of Gn, p is o1, then 1 can be strengthened to ex Gn, p, H 1 + o1 n 2 p Indeed, one can remove all copies of H from Gn, p by arbitrarily removing an edge from each copy of some subgraph H of H and the assumption on p implies that whp only a tiny proportion of the edges will be removed this way Such considerations naturally lead to the notion of 2-density of H, denoted by m 2 H, which is defined by { } eh 1 m 2 H = max v H 2 : H H, e H 2 Moreover, we say that H is 2-balanced if H itself is one of the graphs achieving the maximum above, that is, if m 2 H = e H 1/v H 2 It is straightforward to verify that the expected number of copies of some subgraph H of H that contain a given edge of Gn, p tends to zero precisely when p n 1/m 2H Haxell, Kohayakawa, Luczak, and Rödl [17, 23] conjectured that if the opposite inequality p n 1/m 2H holds, then the converse of 1 must essentially be true The case when H is bipartite is much more subtle; see, eg, [22, 27] This conjecture was proved by Conlon and Gowers [7], under the additional assumption that H is 2-balanced, and independently by Schacht [31]; see also [6, 8, 12, 29, 30] Theorem 12 [7],[31] For any fixed graph H with at least two edges, the following holds whp ex Gn, p, H n 1 1 χh 1 = + o1 2 p if p n 1/m 2 H, 1 + o1 n 2 p if n 2 p n 1/m2H The second generalization of the Turán problem is to fix two graphs T and H and ask to determine the maximum number of copies of T in an H-free subgraph of K n Denote this function by exn, T, H and note that exn, H = exn, K 2, H, so this is indeed a generalization Erdős [9] resolved this question in the case when both T and H are complete graphs, proving that the balanced complete χh 1-partite graph has the most copies of T Another notable result was recently obtained by Hatami, Hladký, Kráľ, Norine, and Razborov [16] and, independently, by Grzesik [14], who determined exn, C 5, K 3, resolving an old conjecture of Erdős The systematic study of the function exn, T, H for general T and H, however, was initiated only recently by Alon and Shikhelman [3] Determining the function exn, T, H asymptotically for arbitrary T and H seems to be a very difficult task and a generalization of Theorem 11 to this broader context has yet to be discovered On the positive side, a nowadays standard argument can be used to derive the following generalization of the Erdős Stone theorem to the case when T is a complete graph from the aforementioned result of Erdős 2

3 Theorem 13 For any fixed nonempty graph H and any integer m 2, χh 1 n m exn, K m, H = + on m m χh 1 Analogously to Theorem 11, in the case χh m, the above theorem only tells us that exn, K m, H = on m The following simple proposition generalizes this fact A blow-up of a graph T is any graph obtained from T by replacing each of its vertices with an independent set and each of its edges with a complete bipartite graph between the respective independent sets Proposition 14 [3] Let T be a fixed graph with t vertices Then exn, T, H = Ωn t if and only if H is not a subgraph of a blow-up of T Otherwise, exn, T, H n t c for some c > 0 that depends only on T and H We remark that both the problems of i determining the limit of exn, T, H n t for general T and H such that H is not contained in a blow-up of T and ii computing exn, T, H up to a constant for arbitrary T and H such that H is contained in a blow-up of T seem extremely difficult Even the case T = K 2 of ii alone, that is, determining the order of magnitude of the Turán function exn, H for an arbitrary bipartite graph H is a notorious open problem, see [13] The common generalization of Theorems 12 and 13 was considered in [2] Let ex Gn, p, T, H be the random variable that counts the maximum number of copies of T in an H-free subgraph of Gn, p Generalizing 1, one can show that the inequality ex Gn, p, T, H exn, T, H + o n v T holds whenever p n vt /et for every nonempty T T It seems natural to guess that the opposite inequality holds as soon as p n 1/m 2H The case T = K m was studied in [2], where the following generalization of Theorem 12 was proved Theorem 15 [2] Let m 2 be an integer and let H be a fixed graph with m 2 H > m 2 K m and χh > m If p is such that n m p m 2 tends to infinity with n, then whp ex Gn, p, K m, H 1 + o1 χh 1 m n m = χh 1 p m 2 if p n 1/m2H, 1 + o1 n m p m 2 if p n 1/m2H Let us draw the reader s attention to the assumption that m 2 H > m 2 K m in the statement of the theorem No such assumption was explicitly present in the statement of Theorem 12 and it is natural to wonder whether it is really necessary Since we assume that H is not m-colorable, then it must contain a subgraph whose average degree is at least m, larger than the average degree of K m In particular, it is natural to guess that this implies that the 2-density of H is larger than the 2-density of K m Perhaps surprisingly, this is not true and only the weaker inequality m 2 H > m 2 K m 1 does hold for every non-m-colorable graph H A construction of a graph H such that χh = 4 and m 2 H < m 2 K 3 was given in [1] Subsequently, constructions of graphs H such that χh = m + 1 but m 2 H < m 2 K m were given for all m in [2] It was also shown there that for such graphs H, the typical value of ex Gn, p, K m, H does not change at p = n 1/m2H, as in Theorem 15 More precisely, if p = n 1/m2H+δ for some small but fixed δ > 0, then still ex Gn, p, K m, H = 1+o1 n m p m 2 This led to the following open questions: 3

4 i Where does the phase transition of ex Gn, p, K m, H take place if m 2 H m 2 K m? ii How does the function p ex Gn, p, T, H grow for general T and H? In this paper we answer both of these questions under the assumptions that T is 2-balanced and H is not contained in a blow-up of T Answering question ii in the case when H is contained in a blow-up of T seems extremely challenging, as even the order of magnitude of exn, T, H, which corresponds to setting p = 1 above, is not known for general graphs T and H, see the comment below Proposition 14 The case when m 2 H > m 2 T holds no surprises, as the following extension of Theorem 15 is valid We denote by N T K n the number of copies of a graph T in the complete graph K n Theorem 16 Suppose that H and T are fixed graphs such that T is 2-balanced and that m 2 H > m 2 T If p n v T /e T, then whp ex Gn, p, T, H { NT K n + o n v T p e T if p n 1/m2H, = ex n, T, H + o n v T p e T if p n 1/m2H As already hinted at by [2], the case when m 2 H m 2 T exhibits a more complex behavior We find that there are several potential phase transitions and we relate their locations to a measure of density of various coverings of H with copies of T that generalizes the notion of the 2-density of H Moreover, we show that the typical asymptotic value of ex Gn, p, T, H is determined, for every p that does not belong to any of the constantly many phase transition windows, by a solution of a deterministic hypergraph Turán-type problem Unfortunately, we were not only unable to solve this Turán-type problem in full generality, but also we do not understand it sufficiently well to either show that for some pairs of T and H, the function p ex Gn, p, T, H undergoes more than one phase transition or to rule out the existence of such pairs We leave these questions as a challenge for future work In order to make the above discussion formal and state the main theorem, we will require several definitions 11 Notations and definitions A T -covering of H is a minimal 2 collection F = {T 1,, T k } of pairwise edge disjoint copies of T in a large complete graph whose union contains a copy of H Given two T -coverings F = {T 1,, T k } and F = {T 1,, T k }, a map f from the union of the vertex sets of the T is to the union of the vertex sets of the T i s is an isomorphism if it is a bijection and for every T i F, the graph ft i belongs to F We can then say that the type of a T -covering of H is just the isomorphism class of this covering Observe that there are only finitely many types of T -coverings of H One special type of a T -covering of H that will be important in our considerations is the covering of H with e H copies of T such that each copy of T intersects H in a single edge and is otherwise completely vertex disjoint from the remaining e H 1 copies of T that constitute this covering We denote this covering by F e and note that the union of all members of F e is a graph with v H + e H v T 2 vertices and e H e T edges 2 The collection F = {T 1,, T k } is minimal in the sense that for every i [k], the union of all graphs in F \ {T i} no longer contains a copy of H 4

5 For a collection F of copies of T, denote by UF the underlying graph of F, that is, the union of all members of F We define the T -density of a T -covering F, which we shall denote by m T F, as follows: { } euf m T F = max e T : F F, F 2 v UF v T Note that this generalizes the notion of 2-density of a graph Indeed, the 2-density of H is the K 2 - density of the edge set of H The notion of T -density is motivated by the following observation Remark 17 For every collection F of at least two copies of T, E [ N UF Gn, p ] E [ NT Gn, p ] for some F F p n 1/m T F For graphs G and T, we let T G denote the collection of copies of T in G and let N T G = T G Even though we are interested in maximizing N T G in an H-free subgraph G K n, we shall be considering more general abstract collections of T -copies in K n that do not contain a T -covering of H of a certain type or a set of types In particular, if G K n is H-free, then T G is one such collection of T -copies, as it does not contain any T -covering of H since the underlying graph of every T -covering of H contains H as a subgraph However, not all the collections we shall consider will be graphic, that is, of the form T G for some graph G The aforementioned Turán-type problem for hypergraphs asks to determine the following quantity For a given family F of T -coverings of H, we let ex n, T, F be the maximum size of a collection of copies of T in K n that does not contain any member of F Note that for any collection F of T -coverings of H, one has that ex n, T, F exn, T, H Indeed, if G is an H-free graph with n vertices such that exn, T, H = N T G, then T G is F-free However, this inequality can be strict, as not every collection of T -copies is of the form T G for some graph G Having said that, we shall show in Lemma 35 that at least ex n, T, F e = exn, T, H + on v T We are now equipped to formulate the key definition needed to state our main result Definition 18 Suppose that T and H are fixed graphs and assume that T is 2-balanced The T -resolution of H is the sequence F 1,, F k of all types of T -coverings of H whose T -density does not exceed m T F e, ordered by their T -density with ties broken in some canonical way The associated threshold sequence is the sequence p 0, p 1,, p k, where p 0 = n v T /e T and p i = n 1/m T F i for i [k] 12 Statment of the main theorem The following theorem is the main result of this paper Theorem 19 Suppose that H and T are fixed graphs and assume that T is 2-balanced and that m 2 H m 2 T Let F 1,, F k be the T -resolution of H and let p 0, p 1,, p k be the associated threshold sequence Then the following holds for every i [k], i If p 0 p p i, then whp ex Gn, p, T, H ex n, T, {F 1,, F i 1 } + o n v T 5

6 ii If p p i, then whp ex Gn, p, T, H ex n, T, {F 1,, F i } + o n v T Even though the above theorem determines the typical values of ex Gn, p, T, H for almost all p, these values remain somewhat of a mystery as we do not know how to compute ex n, T, {F 1,, F i } in general One thing that we do know how to prove is that ex n, T, F = exn, T, H + on v T for every family F of T -coverings of G that contains the special covering F e, see Lemma 35 Moreover, it is not hard to verify that m T F e = e T v T 2 + 1/m 2 H, which, when T is 2-balanced, is equal to the so-called asymmetric 2-density of T and H, a quantity that arises in the study of asymmetric Ramsey properties of Gn, p, see [15, 21, 24] Note that if T is 2-balanced and m 2 H < m 2 T, then m 2 H < m T F e < m 2 T An abbreviated version of Theorem 19 can be now stated as follows Corollary 110 Suppose that H and T are fixed graphs and assume that T is 2-balanced and that m 2 H m 2 T There is an integer k 1 and rational numbers µ 0 < < µ k, where { } et µ 0 = max : T e T T and µ k v T v T 2 + 1/m 2 H, and real numbers π 0 > > π k, where such that whp π 0 = ex Gn, p, T, H = 1 AutT and π k = lim n exn, T, H n v T, { πi + o1n v T, n 1/µ i p n 1/µ i+1 for i {0,, k 1}, π k + o1n v T, p n 1/µ k A rather disappointing feature of Corollary 110 and thus of Theorem 19 is that we are unable to determine whether or not there exists a pair of graphs H and T for which the typical value of ex Gn, p, T, H undergoes more than one phase transition that is, the integer k from the statement of the corollary is strictly greater than one If one was allowed to replace H with a finite family of forbidden graphs, then one can see an arbitrary finite number of phase transitions even in the case when T = K 2, see [26, Theorem 64] Even though we were able to construct pairs of H and T which admit T -coverings of H whose T -density is strictly smaller than the T -density of the special covering of H with e H copies of T, for no such T -covering F we were able to show that ex n, T, F exn, T, H + Ωn v T On the other hand, if one removes the various important assumptions on the densities of H, T, and F, then one can find such triples A simple example is H = K 7, T = K 3, and F being a decomposition of K 7 into edge-disjoint triangles the Fano plane We thus pose the following question Question 111 Do there exist pairs of graphs H and T such that m 2 H m 2 T, T is 2-balanced, and the family F of all T -coverings of H that have the smallest T -density among all T -coverings of H satisfies ex n, T, F exn, T, H + Ωn v T? 6

7 13 Structure of the paper The rest of the paper is structured as follows In Section 2 we give a high level overview of the proofs of Theorems 16 and 19 In Section 3 we introduce the main tools, such as the hypergraph container lemma, and prove a few useful lemmas and corollaries concerning extremal and random graphs In Section 4 we give the proofs of the main theorems, starting with the simpler Theorem 16 and then continuing to Theorem 19 Finally, in Section 5 we give concluding remarks and offer open problems 2 Proof outline Before diving into the details of the proofs of Theorems 16 and 19, let us briefly go over the main steps we take, highlighting the main ideas The proofs of the lower bounds on ex Gn, p, T, H are rather standard Suppose that G Gn, p In the setting of Theorem 16, the H-free subgraph of G with a large number of copies of T is obtained by arbitrarily removing from G one edge from every copy of some subgraph of H In the setting of Theorem 19, we remove from G all edges that are either i not contained in a copy of T that belongs to a fixed extremal {F 1,, F i 1 }-free collection T T K n, or ii contained in a copy of T that constitutes some T -covering of H in T G, or iii contained in more than one copy of T in G Note that all copies of H in G are removed this way Our assumption on p guarantees that in steps ii and iii above we lose only a negligible proportion of T G The proofs of the upper bounds utilize the method of hypergraph containers [6, 30]; see also [5] Suppose that G Gn, p and let G 0 be an H-free subgraph of G In the setting of Theorem 16, a standard application of the method yields a collection C of exp On 2 1/m2H log n containers subgraphs of K n, each with merely on v H copies of H that cover the family of all H-free subgraphs of K n ; in particular, G 0 has to be a subgraph of one of the containers A standard supersaturation result states that each graph in C can have at most exn, T, H+on v H copies of H It follows that for each fixed container C C, the intersection of G with C can have no more than exn, T, H + on v T copies of T At this point, one would normally take the union bound over all containers and conclude that whp the number of copies of T in G C is small simultaneously for all C C and hence also G 0 has this property, as G 0 G C for some C C Unfortunately, we cannot afford to take such a union bound as the rate of the upper tail of the number of copies of T in Gn, p is much too slow to allow this, see [20] Luckily, the rate of the lower tail of the number of copies of T in Gn, p is sufficiently fast, see Lemma 37, to allow a union bound over all containers Therefore, what we do is first prove that whp N T G = 1 + o1n T K n and then show that whp the number of copies of T in G that are not fully contained in C is at least N T K n exn, T, H on v T simultaneously for all C C This implies that whp N T G 0 max C C N T G C exn, T, H + on v T In the setting of Theorem 19, instead of building containers for all possible graphs G 0, we build containers for all possible collections T G 0, exploiting the fact that T G 0 cannot contain any T -covering of H, as G 0 is H-free More precisely, we work with hypergraphs H 1,, H i, each with vertex set T K n, whose edges are copies of the T -coverings F 1,, F i, respectively A version of the container theorem presented in Corollary 32 provides us with a small collection C of subsets of T K n such that i each {F 1,, F i }-free collection T T K n is contained in some member 7

8 of C and ii each C C has only on v UF j copies of each F j and thus by a standard averaging argument it comprises at most ex n, T, {F 1,, F i } copies of T The key parameter q from the statement of Corollary 32, which determines the size of C, exactly matches our definitions of m T F 1,, m T F i Now, since the underlying graph of each F j contains H as a subgraph and G 0 is H-free, T G 0 must be contained in some member of C The rest of the argument is similar to the proof of Theorem 16 we first bound the upper tail of N T G and then the lower tail of T G \ C for all C C simultaneously 3 Tools and Preliminary Results 31 Hypergraph container lemma The first key ingredient in our proof is the following version of the hypergraph container lemma, proved by Balogh, Morris, and Samotij [6] An essentially equivalent statement was obtained independently by Saxton and Thomason [30] We first introduce the relevant notions Suppose that H is a uniform hypergraph For a set B V H, we let deg H B = {A EH : B A} and for each l [k], we let l H = max { deg H B : B V H, B = l } We denote by IH the collection of independent sets in H Theorem 31 [6] For every positive integer k and all positive K and ε, there exists a positive constant C such that the following holds Let H be a k-uniform hypergraph and assume that q 0, 1 satisfies l 1 eh l H Kq for all l [k] 2 vh There exist a family S V CqvH and functions f : S PV H and g : IH S such that: i For every I IH, gi I and I \ gi fgi ii For every S S, eh[fs] εeh iii If gi I and gi I for some I, I IH, then gi = gi Let us remark here that the final assertion of the statement of Theorem 31 is not present in the original statement of [6, Theorem 22] It is, however, proved in the final claim of the proof of [6, Theorem 22] Since the hypergraphs we shall be working with in the proof of Theorem 19 are not necessarily uniform, we shall be actually invoking the following rather straightforward corollary of Theorem 31 Corollary 32 For all positive integers k 1,, k i and all positive K and ε, there exists a positive constant C such that the following holds Suppose that H 1,, H i are hypergraphs with the same vertex set V and that H j is k j -uniform, for each j [i] Assume that q 0, 1 is such that for all j [i], l H j Kq l 1 eh j for all l [k j ] 3 vh j There exist a family S V Cq V and functions f : S PV and g : i j=1 IH j S such that: 8

9 i For every I i j=1 IH j, gi I and I \ gi fgi ii For every S S, eh j [fs] εeh j for every j [i] iii If gi I and gi I for some I, I i j=1 IH j, then gi = gi Proof For each j [i], let C j be the constant given by Theorem 31 with k k j Assume that q 0, 1 is such that the hypergraphs H 1,, H i satisfy 3 For each j [i], we may apply Theorem 31 to the hypergraph H j to obtain a family S j V C j V and functions fj : S j PV and g j : IH j S j as in the assertion of the theorem We now let C = C C i and define S = { S 1 S i : S j S j for each j [i] } V Cq V and, given an I i j=1 IH j, gi = g 1 I g i I Suppose that gi I and gi I for some I, I i j=1 IH j Then also g j I gi I and, similarly, g j I gi I for each j [i] Assertion iii of Theorem 31 implies that that g j I = g j I for each j and thus gi = gi Since gi I, we may also conclude that if gi = gi, then also g j I = g j I for each j [i] In particular, we may define, for each I i j=1 IH j, fgi = f 1 g 1 I f i g i I It is routine to verify that I \gi fgi and that e H j [fgi] εeh j for every j [i] 32 Supersaturation results The following two statements can be proved using a standard averaging argument in the spirit of the classical supersaturation theorem of Erdős and Simonovits [10] Lemma 33 Given graphs H and T and a δ > 0, there exists an ε > 0 such that the following holds Every n-vertex graph G with N T G exn, T, H + δn v T contains more than εn v H copies of H Lemma 34 Given graphs H and T, a finite family F of T -coverings of H, and a δ > 0, there exists an ε > 0 such that the following holds For every collection T T K n with T ex n, T, F, there exists an F F such that T contains more than εn v UF copies of F Our final lemma states that the extremal function ex n, T, F corresponding to a family F of T -coverings of H can be approximated by exn, T, H at least when F contains the special T -covering F e of H with e H copies of T Lemma 35 Given graphs H and T, let F e be the T -covering of H with e H copies of T defined in Section 11 Then ex n, T, F e = exn, T, H + on v T 9

10 Proof Since the underlying graph of F e contains a copy of H, then ex n, T, F e N T G for every H-free graph G This shows that ex n, T, F e exn, T, H For the opposite inequality, fix an arbitrary ε > 0 and suppose that T is a collection of exn, T, H + εn v T copies of T in K n Let E be the set of all edges of K n that belong to fewer than εn v T 2 copies of T from T and let T comprise only those copies of T from T that contain no edge from E Observe that T T E εn v T n 2 T εn v T 2 > exn, T, H 2 Let G K n be the union of all copies of T in T Since N T G T > exn, T, H, the graph G contains a copy of H As each edge of G is contained in at least εn v T 2 copies of T from T, each copy of H in G must be covered by a copy of F e that is contained in T Indeed, given a copy of H in G, one may construct such an F e greedily by considering the edges of H ordered arbitrarily as f 1,, f eh and then finding some T i T that contains f i and whose remaining v T 2 vertices lie outside of V T 1 V T i 1, for each i {1,, e H } in turn One is guaranteed to find such a T i since the number of copies of T in T K n that contain f i and have at least one more vertex in V T 1 V T i 1 is only On v T 3 33 Small subgraphs in Gn, p Our proofs will require several properties of the distribution of the number of copies of a given fixed graph T in the random graph G n,p Following the classical approach of Ruciński and Vince [28], we first prove that if T is 2-balanced, then the number of copies T in G n,p is concentrated around its expectation, provided that this expectation tends to infinity with n Moreover, we show that if p n 1/m 2T, then copies of T in G n,p are essentially pairwise edge disjoint Lemma 36 Suppose that T is a fixed 2-balanced graph, assume that p n v T /e T, and let G Gn, p Then whp N T G = 1 + o1 E[N T G] Moreover, if p n 1/m 2T, then whp G contains a subgraph G such that: i N T G = 1 + o1 E[N T G] ii Every edge of G belongs to exactly one copy of T Proof Assume that p n v T /e T and let G Gn, p Let X = N T G and write Y for the number of pairs of distinct copies of T in G that share at least one edge A routine calculation see, eg, [19, Chapter 3] shows that { VarX E[X] + E[Y ] and E[Y ] C E[X] 2 min n vt p et } 1 : T T for some constant C that depends only on T Since E[X] = Θ n v T p e T, our assumption on p implies that E[X] and, by Lemma 310, that VarX E[X] 2 This proves the first assertion of the lemma To see the second assertion, suppose further than p n 1/m 2T A moment s thought tells us that Lemma 310 actually implies that then { min n vt p et } : T T n v T 10

11 This means, in particular, that E[Y ] E[X] and thus whp Y X Finally, observe that if X = 1 + o1e[x] and Y X, then one may obtain a graph G with the claimed properties by first removing from G all edges that belong to more than one copy of T and subsequently removing all edges that are not contained in any copy of T The following optimal tail estimate for the number of copies of a fixed graph T from a given family T T K n that appear in Gn, p is a rather straightforward extension of the result of Janson, Luczak, and Ruciński [18] Lemma 37 For every graph T and constant δ > 0, there exists a constant β > 0 such that the following holds For every p and each collection T of copies of T in K n, T Pr T Gn, p T δn v T p e T exp β min { n v T : T T } We shall derive Lemma 37 from the following well-known inequality see, for example, [4, Chapter 8] Theorem 38 Janson s Inequality Suppose that Ω is a finite set and let B 1,, B k be arbitrary subsets of Ω Form a random subset R Ω by independently keeping each ω Ω with probability p ω [0, 1] For each i [k], let X i be the indicator of the event that B i R Let X = i X i and define k µ = E[X] = E[X i X j ] = i=1 Then for any 0 t µ, ω B i p ω and = i j B i B j Pr X µ t exp t 2 2µ + i j B i B j ω B i B j p ω Proof of Lemma 37 Suppose that T = {T 1,, T k } and for each i [k], let X i be the indicator of the event that T i appears in Gn, p, so that X = k X i = T T Gn, p i=1 Let µ = E[X] and be as in the statement of Theorem 38 and observe that and that = k i=1 j i T i T j µ = T n v T Pr T i T j Gn, p T =T T 2 e T n 2v T p 2eT min { n v T : T T } 1 It thus follows from Theorem 38 that Pr X µ δn v T p e T exp δ2 n 2v T p 2e T 2µ + as claimed n v T v T p 2e T e T { } exp δ 2 n 2v T 1 p 2eT min 4µ, 1 4 exp 2 e T 2 δ 2 min { n v T : T T }, 11

12 34 Properties of graph densities Finally, we establish several useful facts relating the three notions of graph density that we consider in this work the density, the 2-density, and the T -density Our first lemma partially explains why the two cases m 2 H m 2 T and m 2 H > m 2 T that we consider separately while studying the typical value of ex Gn, p, T, H are so different Lemma 39 Suppose that H and T are fixed graphs and assume that T is 2-balanced i If m 2 H m 2 T, then the T -covering F e of H with e H edges satisfies m T F e m 2 T ii If m 2 H > m 2 T, then every T -covering F of H satisfies m T F > m 2 T Proof To see i, assume that m 2 H m 2 T and fix some F F e Since F is a T -covering of some subgraph H H with F edges, then e UF e T v UF v T = e H e T e T = e H 1e T 1 + e H 1 v H + e H v T 2 v T e H 1v T 2 + v H 2 { et 1 max v T 2, e H 1 } max{m 2 T, m 2 H} = m 2 T v H 2 To see ii, assume that m 2 H > m 2 T and let F be an arbitrary T -covering of H Let H H be any subgraph of H satisfying e H 1 v H 2 > e T 1 v T 2 and denote by T 1,, T k all those elements of F that intersect H For each i [k], denote by v i and e i the numbers of vertices and edges of T i H, respectively, and note that e i 1 m 2 T v i 2 One easily verifies that m T F e UF e T = e H + k i=1 e T e i e T v UF v T v H + k i=1 v T v i v T as claimed = e H 1 + k 1e T 1 k i=1 e i 1 v H 2 + k 1v T 2 k i=1 v i 2 > e T 1 v T 2 = m 2T, Our next lemma computes the rate of the lower tail of the number of copies of a 2-balanced graph T in Gn, p, which Lemma 37 provides in a somewhat implicit form Lemma 310 If T is a 2-balanced graph, then { { min n vt p et } n : T v T if p n 1/m 2T, T = n 2 p if p n 1/m 2T Proof Let T be a 2-balanced graph Suppose first that p n 1/m 2T and fix some T T with at least two edges Since T is 2-balanced, then m 2 T e T 1 v T 2 v T 2 e and hence p n T 1 It follows that n v T = n 2 p n v T 2 1 n 2 p n v T 2 n v T 2 et 1 e T 1 = n 2 p 12

13 Suppose now that p n 1/m 2T and fix a nonempty T T Since m 2 T e T 1 v T 2, then e T e T m 2 T = e T 1 e T 1 m 2 T v T 2 v T 2 = v T v T It follows that n v T = n v T p et n v T v T e T n v T p et n v T v T n 1/m e 2T T e T n v T 4 Proof of Theorems 16 and Proof of Theorem 16 Suppose that H and T are fixed graphs and assume that T is 2-balanced and that m 2 H > m 2 T Proof of the 0-statement Suppose that n v T /e T p n 1/m2H and let G Gn, p Our aim is to show that for every positive constant δ, whp G contains an H-free subgraph with at least NT K n δn v T p e T copies of T We shall argue somewhat differently depending on whether or not p n 1/m 2T Case 1 p n 1/m 2T Suppose that G satisfies both assertions of Lemma 36 and let G be the subgraph of G from the statement of the lemma Since each edge of G is contained in exactly one copy of T, then each copy of H in G must correspond to some T -covering of H in T G 3 Consider an arbitrary T -covering F of H Since we have assumed that m 2 H > m 2 T, part ii of Lemma 39 yields m T F > m 2 T Since p n 1/m2H n 1/m T F, there is some F F such that E [ N UF G ] E [ N T G ] Since there are only O1 types of T -coverings of H, then whp one may remove from G some o E[N T G] edges to obtain an H-free graph G 0 Since each edge of G belongs to exactly one copy of T, then N T G 0 = N T G o E[N T G] 1 + o1 N T K n Case 2 p = Ωn 1/m 2T Suppose that G satisfies the assertion of Lemma 36 Since p n 1/m 2H, then there is some H H such that E[N H G] n 2 p In particular, whp one may delete at most on 2 p edges from G to make it H-free It suffices to show that whp for every set X EG, the graph G \ X contains at least N T K n δn v T p e T copies of T For a fixed X EK n, let A X denote the event that N T G \ X N T K n δn v T Since X n 2, then N T K n \ X = N T K n on v T and thus Lemma 37 with T T K n \ X together with Lemma 310 yield PrA X exp βn 2 p 3 Recall that T -coverings of H are collections of pairwise edge disjoint copies of T 13

14 for some positive constant β Since for every X EK n, the event X G is increasing and the event A X is decreasing, Harris inequality implies that Consequently, Pr A X for some X G with X = on 2 p x n 2 p n 2 x Pr X G and A X Pr X G PrAX p x exp βn 2 p X EK n X n 2 p x n 2 p as the function x ea/x x is increasing when x a en 2 p 2x p x exp βn 2 p x exp βn 2 p exp βn 2 p/2, Proof of the 1-statement Suppose that p n 1/m 2H and let G Gn, p Our aim is to show that for every positive constant δ, whp every H-free subgraph G 0 of G satisfies N T G 0 exn, T, H + 2δn v T Let H be the e H -uniform hypergraph with vertex set EK n whose edges are all copies of H in K n Observe that vh = Θ n 2 and eh = Θ n v H Observe that there is a natural one-to-one correspondence between the independent sets of H and H-free subgraphs of K n As we shall be applying Theorem 31 to the hypergraphs H, we let q = n 1/m2H and verify that H satisfies the main assumption of the theorem, provided that K is a sufficiently large constant Claim 41 There is a constant K such that the hypergraph H satisfies 2 in Theorem 31 with q = n 1/m 2H Proof Fix an arbitrary l [k] and observe that l H is the largest number of copies of H in K n that contain some given set of l edges It follows that l H n v H v H and hence H H,e H =l vh eh max l H n v H v H l [k] q l 1 2 eh n 2 vh max =H H q e H 1 = 2 eh 1 min =H H nv H 2 q e H 1 Finally, since q = n 1/m 2H, then n v H 2 q e H 1 1 for every nonempty H H Denote by Free n H the family of all H-free subgraphs of K n Apply Theorem 31 to the hypergraph H to obtain a constant C, a family S EK n Cqn, and functions g : 2 Freen H S and f : S PEK n such that: i For every G 0 Free n H, gg 0 G 0 and G 0 \ gg 0 fgg 0 14

15 ii For every S S, the graph fs contains at most εn v H copies of H Given an S S, denote by A S the event T G \ T fs S N T K n exn, T, H δn v T Claim 42 There is a constant β > 0 such that for every S S, PrA S exp βn 2 p Proof Fix an S S and let T S denote the collection of all copies of T in K n that are not completely contained in fs S Since S n 2, then property ii above and Lemma 33 imply that T S = N T K n N T fs S NT K n N T fs S n v T 2 N T K n exn, T, H δn v T /2 Since T is 2-balanced and p n 1/m 2H n 1/m 2T, Lemmas 37 and 310 imply that PrA S exp β min { n v T : T T } = exp βn 2 p for some positive constant β, as claimed Suppose that G satisfies the assertion of Lemma 36 and let G 0 G be an H-free subgraph of G that maximizes N T G 0 Since G 0 Free n H, then and hence gg 0 G 0 fgg 0 gg 0 N T G 0 max { T G T fs S : S S and S G } = N T G min { T G \ T fs S : S S and S G } = 1 + o1 E[N T G] min { T G \ T fs S : S S and S G } We shall show that whp A S does not hold for any S S such that S G, which will imply that N T G 0 exn, T, H + 2δn v T Since for every S S, the event S G is increasing and the event A S is decreasing, Harris inequality implies that Pr S G and A S Pr S G PrAS By Claim 42, in order to complete the proof in this case, it is sufficient to prove the following Claim 43 Pr S G exp on 2 p S S Proof Since each S S is a graph with at most Cqn 2 edges and q = n 1/m2H p, then Pr S G n 2 p s en 2 s p = exp o n 2 p, s s S S s Cqn 2 as the function s ea/s s is increasing when s a s=opn 2 This completes the proof of the 1-statement in Theorem 16 15

16 42 Proof of Theorem 19 Suppose that H and T are fixed graphs and assume that T is 2-balanced and that m 2 H m 2 T Recall Definition 18, let F 1,, F k be the T -resolution of H, and let p 0, p 1,, p k be the associated threshold sequence For each i {0,, k}, denote by F i the set {F 1,, F i } Proof of part i Fix an i [k], suppose that p 0 p p i, and let G Gn, p Our aim is to show that for every positive constant δ, whp G contains an H-free subgraph with at least ex n, T, F i 1 δn v T p e T copies of T If ex n, T, F i 1 = o n v T, then the assertion is trivial we may simply take the empty graph, so for the remainder of the proof we shall assume that ex n, T, F i 1 γn v T for some positive constant γ It follows from part i of Lemma 39 that p p i n 1/m T F e n 1/m 2T, so we may assume that G satisfies both assertions of Lemma 36 Let G be the subgraph of G from the statement of the lemma and let T i 1 be an extremal collection of copies of T in K n with respect to being F i 1 -free In other words, let T i 1 be a collection of ex n, T, F i 1 copies of T that does not contain any T -covering of either of F 1,, F i 1 Let G be the graph obtained from G by keeping only edges covered by T G T i 1 and let G 0 be the graph obtained from G by deleting all edges from every copy of H in G This graph is clearly H-free Since each edge of G is contained in exactly one copy of T, then each copy of H in G must belong to some T -covering of H Since T i 1 is F i 1 -free, then the only T -coverings of H that we may find in G are F i,, F k and coverings whose T -density is strictly greater than m T F e Since p p i n 1/m T F e and there are only O1 types of T -coverings, then whp there are only o E[N T G] edges in G \ G 0 and thus N T G N T G 0 = o E[N T G], as every edge of G belongs to at most one copy of T Now, Lemmas 37 and 310 imply that whp Therefore, T G T i 1 ex n, T, F 1 δn v T /3 N T G 0 N T G δn v T /3 = T G T i 1 δn v T /3 T G T i 1 N T G N T G δn v T /3 ex n, T, F 1 δn v T p e T, since N T G = N T G + o n v T Proof of part ii Fix an i [k], suppose that p p i, and let G Gn, p Our aim is to show that for every positive constant δ, whp every H-free subgraph G 0 of G satisfies N T G 0 ex n, T, F i + δn v T p e T 4 For each j [i], let H j be the F j -uniform hypergraph whose vertices are all copies of T in K n and whose edges are all collections of F j copies of T in K n that are isomorphic to the T -covering F j Observe that vh j = Θ n v T and eh j = Θ n v UF j Since UF j contains a copy of H, then for every H-free graph G 0, the family T G 0 is an independent set in H j, for each j [i] As we shall be applying Corollary 32 to the hypergraphs H 1,, H i, we let q = i and verify that all H j satisfy the main assumption of the corollary, provided that K is a sufficiently large constant 16

17 Claim 44 There is a constant K such that for each j [i], the hypergraph H j satisfies 3 in Corollary 32 with q = i Proof Fix an arbitrary l [k] and observe that l H j is the largest number of copies of F j in T K n that share the same set of l copies of T It follows that l H j n v UF j v UF and hence vh j eh j max l H j l [k] q l 1 2 F j Finally, since p i p j = n 1/m T F j, then for every nonempty F F j F F j, F =l n v T n v max UF j = 2 F j n vufj vuf =F F j F 1 i min n v UF v T p e UF e T =F i F j n v UF v T p e UF e T i 1 1 Denote by Free n F i the family of all subfamilies of T K n that do not contain any T -covering isomorphic to one of the members of F i Apply Corollary 32 to the hypergraphs H 1,, H i to obtain a constant C, a family S T K n Cqn v T, and functions g : Freen F i S and f : S PT K n such that: i For every T Free n F i, gt T and T \ gt fgt ii For every S S, the collection fs has at most εn v UF j copies of F j for every j [i] iii If gt T and gt T for some T, T Free n F i, then gt = gt Given an S S, denote by A S the event T G \ fs NT K n ex n, T, F i δn v T Claim 45 There is a constant β > 0 such that for every S S, PrA S exp β min { n 2 p, n v T } Proof Fix an S S and let T S denote the collection of all copies of T in K n that do not belong to fs Property ii above and Lemma 34 imply that T S = N T K n fs N T K n ex n, T, F i δn v T /2 Since T is 2-balanced, Lemmas 37 and 310 imply that PrA S exp β min { n v T : T T } = exp β min { n 2 p, n v T } for some positive constant β, as claimed We shall now argue somewhat differently depending on whether or not p n 1/m 2T 17

18 Case 1 p n 1/m 2T Suppose that G satisfies both assertions of Lemma 36 and let G be the subgraph of G from the statement of the lemma Let G 0 G be an H-free subgraph of G that maximizes N T G 0 and let G = G 0 G Since N T G 0 N T G + N T G N T G = N T G + o E[N T G] = N T G + o n v T, it suffices to show that 4 holds with G 0 replaced by G Since G is H-free, then T G Free n F i and hence g T G T G f g T G g T G But this means that N T G max { T G fs + S : S S and S T G } = N T G min { T G \ fs S : S S and S T G } 1 + o1 E[N T G] + C i n v T min { T G \ fs : S S and S T G } = 1 + o1 E[N T G] min { T G \ fs : S S and S T G } Now, let S comprise all the sets of T -copies S S that are pairwise edge disjoint Since T G is a collection of pairwise edge disjoint copies of T, then min { T G \ fs : S S and S T G } = min { T G \ fs : S S and S T G } We shall now show that whp A S does not hold for any S S such that S T G, which will imply that N T G ex n, T, F + 2δn v T Since for every S S, the event S T G is increasing and the event A S is decreasing, Harris inequality implies that Pr S T G and A S Pr S T G PrAS Since we have assumed that p n 1/m 2T, then Claim 45 and Lemma 310 imply that PrA s exp βn v T and consequently, in order to complete the proof in this case, it is sufficient to prove the following Claim 46 S S Pr S T G exp on v T Proof Since each S S consists of pairwise edge disjoint copies of T, then Pr S T G = Pr US G = p e US = S Since S contains only sets of at most C i n v T copies of T in K n and p i p, it now follows that n Pr S T G = p et S v T s s S S S S s C i n v T en v T p e T s = exp o n v T p e T, s s=o n v T since the function s ea/s s is increasing when s a 18

19 Case 2 p = Ωn 1/m 2T Suppose that G satisfies the assertion of Lemma 36 and let G 0 G be an H-free subgraph of G that maximizes N T G 0 Since G 0 is H-free, then T G 0 Free n F i and hence g T G 0 T G 0 f g T G 0 g T G 0 Now, let S comprise all the sets of T -copies S S that are of the form g T G for some H-free graph G K n and observe that N T G 0 max { T G fs + S : S S and S T G } = N T G min { T G \ fs S : S S and S T G } = 1 + o1 E[N T G] min { T G \ fs : S S and S T G } Analogously to Case 1, we shall show that whp A S does not hold for any S S such that S T G, which will imply that N T G 0 ex n, T, F + 2δn v T As before, since for every S S, the event S T G is increasing and the event A S is decreasing, Harris inequality implies that Pr S T G and A S Pr S T G PrAS Since we have assumed that p = Ω n 1/m 2T, then Claim 45 and Lemma 310 imply that PrA s exp βn 2 p and consequently, in order to complete the proof in this case, it is sufficient to prove the following Claim 47 S S Pr S T G exp on 2 p Proof We claim that the function U that maps a collection of copies of T to its underlying graph is injective when restricted to S Indeed, suppose that U g T G 1 = U g T G 2 for some H-free graphs G 1 and G 2 It follows that g T G 1 T U g T G 1 = T U g T G 2 T U T G 2 = T G 2 and, vice-versa, g T G 2 T G 1 The consistency property of the function g, see iii above, implies that g T G 1 = g T G 2 Since p i n 1/m 2T by part i of Lemma 39, then Lemma 310 implies that n v T i n 2 p i In particular, each S S comprises at most Cn 2 p i copies of T and therefore US has at most Ce T n 2 p i edges Since the function U is injective when restricted to S, we may conclude that S T G = US G = Pr U G = p e U S S Pr S S Pr u Ce tn 2 p i n 2 u p u U US s=opn 2 since the function u ea/u u is increasing when u a This completes the proof of part ii of Theorem 19 U US en 2 u p = exp o n 2 p, u 19

20 5 Concluding remarks and open questions In this paper, we have studied the random variable ex Gn, p, T, H that counts the largest number of copies of T in an H-free subgraph of the binomial random graph Gn, p We restricted our attention to the case when T is 2-balanced; the case when T is not 2-balanced poses further challenges and we were not able to resolve it using our methods The threshold phenomena associated with the variable ex Gn, p, T, H are quite different depending on whether or not the inequality m 2 H > m 2 T holds: i If m 2 H > m 2 T, then our Theorem 16 offers a natural generalization of the sparse random analogue of the Erdős Stone theorem that was proved several years ago by Conlon and Gowers [7] and by Schacht [31] ii If m 2 H m 2 T, then the evolution of the random variable ex Gn, p, T, H as p grows from 0 to 1 exhibits a more complex behavior Our Theorem 19 shows that there are several potential phase transitions and that the typical values of the variable between these phase transitions are determined by solutions to deterministic hypergraph Turán-type problems which we were unable to solve in full generality There are several natural directions for further investigations that are suggested by this work: It would be interesting to study the variable ex Gn, p, T, H for general graphs T and H, that is, without assuming that T is 2-balanced We have very little understanding of the Turán-type problems related to T -coverings of H that are described in Section 11, even in the case when T is a complete graph A concrete problem that we found the most interesting is stated as Question 111 In short, we ask if there exists a pair of graphs T and H such that the variable ex Gn, p, T, H undergoes multiple phase transitions Given a family H of graphs, one may more generally ask to study the random variable ex Gn, p, T, H that counts the largest number of copies of T in a subgraph of Gn, p that is free of every H H This problem is solved when T = K 2 and H is finite, see [26, Theorem 64], but not much is known, even in the deterministic case p = 1, when T K 2 References [1] P Allen, J Böttcher, S Griffiths, Y Kohayakawa, and R Morris, The chromatic thresholds of graphs, Adv Math , [2] N Alon, A Kostochka, and C Shikhelman, Many cliques in H-free subgraphs of random graphs, J Comb, to appear [3] N Alon and C Shikhelman, Many T copies in H-free graphs, J Combin Theory Ser B ,

21 [4] N Alon and J H Spencer, The probabilistic method, fourth ed, Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc, Hoboken, NJ, 2016 [5] J Balogh, R Morris, and W Samotij, The method of hypergraph containers, arxiv: [mathco] [6], Independent sets in hypergraphs, J Amer Math Soc , [7] D Conlon and W T Gowers, Combinatorial theorems in sparse random sets, Ann of Math , [8] D Conlon, W T Gowers, W Samotij, and M Schacht, On the K LR conjecture in random graphs, Israel J Math , [9] P Erdős, On the number of complete subgraphs contained in certain graphs, Magyar Tud Akad Mat Kutató Int Közl , [10] P Erdős and M Simonovits, Supersaturated graphs and hypergraphs, Combinatorica , [11] P Erdős and A H Stone, On the structure of linear graphs, Bull Amer Math Soc , [12] E Friedgut, V Rödl, and M Schacht, Ramsey properties of random discrete structures, Random Structures Algorithms , [13] Z Füredi and M Simonovits, The history of degenerate bipartite extremal graph problems, Erdös centennial, Bolyai Soc Math Stud, vol 25, János Bolyai Math Soc, Budapest, 2013, pp [14] A Grzesik, On the maximum number of five-cycles in a triangle-free graph, J Combin Theory Ser B , [15] L Gugelmann, R Nenadov, Y Person, N Škorić, A Steger, and H Thomas, Symmetric and asymmetric Ramsey properties in random hypergraphs, Forum Math Sigma , e28, 47 [16] H Hatami, J Hladký, D Král, S Norine, and A Razborov, On the number of pentagons in triangle-free graphs, J Combin Theory Ser A , [17] P E Haxell, Y Kohayakawa, and T Luczak, Turán s extremal problem in random graphs: forbidding even cycles, J Combin Theory Ser B , [18] S Janson, T Luczak, and A Ruciński, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, Random graphs 87 Poznań, 1987, Wiley, Chichester, 1990, pp [19] S Janson, T Luczak, and A Rucinski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000 [20] S Janson, K Oleszkiewicz, and A Ruciński, Upper tails for subgraph counts in random graphs, Israel J Math ,

22 [21] Y Kohayakawa and B Kreuter, Threshold functions for asymmetric Ramsey properties involving cycles, Random Structures Algorithms , [22] Y Kohayakawa, B Kreuter, and A Steger, An extremal problem for random graphs and the number of graphs with large even-girth, Combinatorica , [23] Y Kohayakawa, T Luczak, and V Rödl, On K 4 -free subgraphs of random graphs, Combinatorica , [24] Y Kohayakawa, M Schacht, and R Spöhel, Upper bounds on probability thresholds for asymmetric Ramsey properties, Random Structures Algorithms , 1 28 [25] T Kővári, V T Sós, and P Turán, On a problem of K Zarankiewicz, Colloquium Math , [26] R Morris, W Samotij, and D Saxton, An asymmetric container lemma and the structure of graphs with no induced 4-cycle, arxiv: [mathco] [27] R Morris and D Saxton, The number of C 2l -free graphs, Adv Math , [28] A Ruciński and A Vince, Balanced graphs and the problem of subgraphs of random graphs, Proceedings of the sixteenth Southeastern international conference on combinatorics, graph theory and computing Boca Raton, Fla, 1985, vol 49, 1985, pp [29] W Samotij, Stability results for random discrete structures, Random Structures Algorithms , [30] D Saxton and A Thomason, Hypergraph containers, Invent Math , [31] M Schacht, Extremal results for random discrete structures, Ann of Math , [32] M Simonovits, Paul Erdős influence on extremal graph theory, The mathematics of Paul Erdős, II, Algorithms Combin, vol 14, Springer, Berlin, 1997, pp