Balanced supersaturation and Turan numbers in random graphs

Balanced supersaturation and Turán numbers in random graphs arXiv:2208.10572v2 [math.CO] 25 Aug 2022 Tao Jiang ∗ Sean Longbrake† August 26, 2022 Abstract In a ground-breaking work utilizing the container method, Morris and Saxton [44] resolved a conjecture of Erdős on the number of C2ℓ -free graphs on n vertices and gave new bounds on the Turán number of C2ℓ in the Erdős-Rényi random graph G(n, p). A key ingredient of their work is the so-called balanced supersaturation property of even cycles of a given length. This motivated Morris and Saxton to make a broad conjecture of the existence of such a property for all bipartite graphs. Roughly speaking, the conjecture states that given a bipartite graph H if an n-vertex graph G has its number of edges much larger than the Turán number ex(n, H) then G contains a collection of copies of H, in which no subset of edges of G are covered more than some naturally expected number of times. In a subsequent breakthrough, Ferber, McKinley, and Samotij [28] established a weaker version of the Morris-Saxton conjecture and applied it to derive far-reaching results on the enumeration problem of H-free graphs. However, this weaker version seems insufficient for applications to the Turán problem for random graphs. Building on these earlier works, in this paper, we essentially prove the conjecture of Morris and Saxton. We show that the conjecture holds when we impose a very mild assumption about H, which is widely believed to hold for all bipartite graphs that contain a cycle. In addition to retrieving the enumeration results of Ferber, McKinley, and Samotij [28], we also obtain some general upper bounds on the Turán number ex(G(n, p), H) of a bipartite graph H in the random graph G(n, p), from which Morris and Saxton’s result on ex(G(n, p), C2ℓ ) would also follow. 1 Introduction Given a graph H, we say that a graph G is H-free if it doesn’t contain H as a subgraph. For a family F of graphs, we say G is F-free if it doesn’t contain any member of F as a subgraph. For a given ∗ Department of Mathematics, Miami University, Oxford, OH 45056, USA. Email: [email protected]. Research supported by National Science Foundation grant DMS-1855542. † Department of Mathematics, Miami University, Oxford, OH 45056, USA. Current address: Department of Mathematics, Emory University, Atlanta, GA 30322, USA. Email: [email protected]. Research partially supported by National Science Foundation grant DMS-1855542. 1 positive integer n and a graph H, the extremal number ex(n, H) denotes the maximum number of edges in an H-free graph n-vertex graph. (For a family F, ex(n, F) is analogously defined for F-free graphs.) A central problem in extremal graph theory is to determine the extremal number ex(n, H) and the typical structure of H-free graphs. Such a study was initiated by Turán [52] in the 1940s, who determined precisely the extremal number of the complete graph. The problem of studying ex(n, H) is therefore also referred to as the Turán problem. Erdős and Stone [27] (see also [23]) determined asymptotically the extremal number ex(n, H) for any non-bipartite graph H, thus leaving estimating ex(n, H) for a bipartite graph H the main remaining challenge in the field. Kővári, Sós and Turán [42] showed that ex(n, Ks,t ) = O(n2−1/s ), where Ks,t denotes the complete bipartite graph with part sizes s and t. This was later shown to be asymptotically tight when t > s! by Kollár Rónyi, and Szabó [43] and when t > (s−1)! by Alon, Rónyai, and Szabó [2]. Both of these were obtained via algebraic constructions. More recently, an innovative random algegraic approach had led to many new tight lower bound constructions, see [11, 12, 13] for instance. In particular, Bukh and Conlon [13] applied the method to settle a long-standing conjecture that asserts that for every rational number α in [1, 2) there is a family F of bipartite graphs for which ex(n, F) = Θ(nα ). A series of recent progresses have also been made on the related exponent conjecture for single bipartite graphs H (rather than forbidding a family F), resulting in many rationals α ∈ [1, 2] and bipartite graphs H for which ex(n, H) = Θ(nα ), see for instance [17, 18, 33, 34, 36, 37]. Another general result is due to Füredi [29], and independently to Alon, Krivelevich, and Sudakov [1] that asserts that ex(n, H) = O(n2−1/s ) for every bipartite graph H where vertices in one part have degree at most s (see [31, 35] for recent generalizations of this result). Alon, Krivelevich, Sudakov [1], in addition, showed that ex(n, H) = O(n2−1/4s ) for every s-degenerate bipartite graph H. Despite these substantial progresses on the bipartite Turán problem, it remains to be the case that for most bipartite graphs H there exist substantial gaps between best known lower and upper bounds on ex(n, H), even for even cycles C2ℓ , where ℓ 6= 2, 3, 5. For more background, the reader is referred to the excellent survey by Füredi and Simonovits [30]. Erdős, Kleitman and Rothschild [21] introduced the problem of counting H-free graphs on n vertices. They showed that there are 2(1+o(1)) ex(n,Kr ) Kr -free graphs, and furthermore that almost all triangle-free graphs are bipartite. Erdős, Frankl and Rödl [20] generalized the former result, showing that for any non-bipartite H there are 2(1+o(1)) ex(n,H) H-free graphs. Kolaitis, Prömel and Rothschild [41] generalized the latter result, showing that almost all Kr -free graphs are (r − 1)-partite. This was further extended by Prömel and Steger [48] to all r-critical graphs. Using hypergraph regularity method, Nagle, Rödl and Schacht [47] showed that there are 2(1+o(1)) ex(n,H) H-free kuniform hypergraphs for any non-k-partite k-uniform hypergraph H. Balogh, Morris and Samotij [7] and Saxton and Thomason [50] reproved this result using the hypergraph container method. Balogh, Bollobás and Simonovits [3, 4, 5] proved more precise counting and structural results for graphs. For bipartite H, Kleitman and Winston [39] made the first breakthrough by showing that there are 2(1+c) ex(n,C4 ) C4 -free graphs on n vertices, where c ≈ 1.17 and resolving a long-standing question of Erdős. No further progress was made until the recent significant work of Balogh and Samotij [8, 9], 2 2−1/s ) ) K -free graphs on n vertices. who showed for every 2 ≤ s ≤ t that there are at most 2(O(n s,t The next major breakthrough was made by Morris and Saxton [44], who showed that the number 1+1/ℓ ) of C2ℓ -free graphs on n vertices is at most 2O(n , confirming a conjecture of Erdős. There are two key ingredients in Morris and Saxton’s work. One is the framework of the container method and the other is the so-called balanced supersaturation property of C2ℓ , stated in one of their main theorems as below. Theorem 1.1 (Morris-Saxton [44]). For every ℓ ≥ 2, there exist constants C > 0, δ > 0 and k0 ∈ N such that the following holds for every k ≥ k0 and every n ∈ N. Given a graph G with n vertices and kn1+1/ℓ edges, there exist a collection H of copies of C2ℓ in G, satisfying: (a) |H| ≥ δk 2ℓ n2 , and |σ|−1 (b) dH (σ) ≤ C · k2ℓ−|σ|− ℓ−1 n1−1/ℓ for every σ ⊂ E(G) with 1 ≤ |σ| ≤ 2ℓ − 1, where dH (σ) = |{A ∈ H : σ ⊂ A}| denotes the ‘degree’ of the set σ in H. Using their container method framework and Theorem 1.1, Morris and Saxton [44] were able to not only obtain the enumeration result on the number of C2ℓ -free graphs on n vertices, but also make further progress on the Turán number of C2ℓ in the Erdős-Rényi random graph G(n, p), where as usual G(n, p) denotes the random graph on [n] where each pair ij is included as an edge independently with probability p. Given a graph H, let ex(G(n, p), H) := max{e(G) : G ⊂ G(n, p) and G is H-free}. Note that both G(n, p) and ex(G(n, p), H) are random variables. The problem of determining the threshold function for the maximum number of edges in an H-free subgraph of G(n, p) has received much attention. For more thorough discussion, the reader is referred to the excellent survey by Rödl and Schacht [49]. The most significant work was the following breakthrough, first independently obtained by Conlon and Gowers [16] (under the assumption that H is strictly 2-balanced, see Definition 1.6) and by Schacht [51], then reproved using the container method by Balogh, Morris and Samotij [7] and independently by Saxton and Thomason [50]. Theorem 1.2 (Conlon-Gowers [16], Schacht [51]). Let H be a graph with ∆(H) ≥ 2 and chromatic number χ(H). Let m2 (H) = max{(e(F ) − 1)/(v(F ) − 2) : F ⊆ H, v(F ) ≥ 3}. For every ε > 0 there exists a constant C > 0 such that if p ≥ Cn−1/m2 (H) , then ex(G(n, p), H) ≤ (1 −   n 1 + ε) p, 2 χ(H) − 1 with high probability, as n → ∞. While Theorem 1.2 give a satisfactory answer for non-bipartite H, much less is known for bipartite H. For C2ℓ , Haxell, Kohayakawa and Luczak [32] showed that if p ≫ n−1+1/(2ℓ−1) then 3 ex(G(n, p), C2ℓ ) ≪ e(G(n, p)), whereas if p = o(n−1+1/(2ℓ−1 ) then ex(G(n, p), C2ℓ ) = (1+o(1))e(G(n, p)). 2 For p = αn−1+1/(2ℓ−1) and 2 ≤ α ≤ n1/(2ℓ−1) , Kohayakawa, Kreuter and Steger [40] obtained the tight result that with high probability,   ex(G(n, p), C2ℓ ) = Θ n1+1/(2ℓ−1) (log α)1/(2ℓ−1) . For recent work on the analogous problem for linear cycles in random r-uniform hypergraphs, see Mubayi and Yepremyan [46]. Applying the container method to Theorem 1.1, Morris and Saxton obtained the following. Theorem 1.3 (Morris-Saxton [44]). For every ℓ ≥ 2, there exists a constant C = C(ℓ) > 0 such that ex(G(n, p), C2ℓ ) ≤ with high probability as n → ∞.  n1+1/(2ℓ−1) (log n)2 p1/ℓ n1+1/ℓ if p ≤ n−(ℓ−1)/(2ℓ−1) · (log n)2ℓ otherwise A well-known conjecture of Erdős and Simonovits (see [30]) states that ex(n, {C3 , C4 , . . . , C2ℓ }) = Θ(n1+1/ℓ ). Morris and Saxton [44] further showed that with high probability as n → ∞, ex(n, G(n, p)) = Ω(p1/ℓ n1+1/ℓ ) for each ℓ for which the Erdős-Simonovits conjecture is true. The successful applications of Theorem 1.1 to both the enumeration problem and the random Turán problem on C2ℓ motivated Morris and Saxton [44] to make a general conjecture about all bipartite graphs. Conjecture 1.4 (Morris-Saxton [44]). Given a bipartite graph H, there exist constants C > 0, ε > 0 and k0 ∈ N such that the following holds. Let k ≥ k0 , and suppose that G is a graph on n vertices with k · ex(n, H) edges. Then there exists a (non-empty) collection H of copies of H in G, satisfying dH (σ) ≤ C · |H| k(1+ε)(|σ|−1) e(G) for every σ ⊂ E(G) with 1 ≤ |σ| ≤ e(H). An important aspect of this conjecture is that its truth would immediately yield desired enumeration results on H-free graphs, in the following sense. Proposition 1.5 (Morris-Saxton [44]). Let H be a bipartite graph. If Conjecture 1.4 holds for H, then there are at most 2O(ex(n,H)) H-free graphs on n vertices. The work of Morris and Saxton and Conjecture 1.4 generated a lot of interest in the field. In a recent breakthrough, Ferber, McKinley and Samotij [28] were able to establish a weaker version of Conjecture 1.4 and applied it to obtain very general enumeration results on H-free hypergraphs for all r-partite r-uniform hypergraphs (r-graphs in short) H that satisfy a very mild assumption which is widely believed to hold for all r-partite r-graphs. The weaker version of Conjecture 1.4 that Ferber, McKinley and Samotij [28] proved is a bit technical to state here and does not seem to 4 immediately apply to the random Turán problem. However, the enumeration results it yields are very general and significant. Definition 1.6 (r-density and proper r-density). Let r ≥ 2 be an integer. Given an r-graph H with v(H) ≥ r + 1, we define its r-density to be mr (H) = max{ e(F ) − 1 : F ⊆ H, v(F ) > r}. v(F ) − r We define its proper r-density to be m∗r (H) = max{ e(F ) − 1 : F ( H, v(F ) > r}. v(F ) − r If mr (H) > m∗r (H), we say that H is strictly r-balanced. 2ℓ−1 and m∗2 (C2ℓ ) = 1. Throughout the For instance, for the even cycle C2ℓ , we have m2 (C2ℓ ) = 2ℓ−2 paper, we will tacitly assume that H has at least two edges. The main results of Ferber, McKinley and Samotij are Theorem 1.7 (Ferber-McKinley-Samotij [28]). Let H be an r-uniform hypergraph and let α and A be positive constants. Suppose that α > r − mr1(H) and that ex(n, H) ≤ Anα for all n. Then α there exists a constant C depending only on α, A, and H such that for all n, there are at most 2Cn H-free r-uniform hypergraphs on n vertices. Theorem 1.8 (Ferber-McKinley-Samotij [28]). Let H be an r-uniform hypergraph and assume that r− 1 +ε ex(n, H) ≥ εn mr (H) for some ε > 0 and all n. Then there exists a constant C depending only α on α, A, and H such that for all n, there are at most 2Cn H-free r-uniform hypergraphs on n vertices. To get a sense of what the condition on H in Theorem 1.8 came from, observe that a simple first moment argument shows that for any r-uniform hypergraph H, ex(n, H) ≥ Ω(nr−1/mr (H) ) holds. In the 2-uniform case, it is widely believed that this simple probabilistic lower bound is not asymptotically tight for any H that contains a cycle. The r ≥ 3 case is expected to be similar. Ferber, McKinley and Samotij made the following conjecture. Conjecture 1.9 (Ferber-McKinley-Samotij [28]). Let H be an arbitrary graph that is not a forest. There exists an ε > 0 such that ex(n, H) ≥ εn2−1/m2 (H)+ε . Conjecture 1.9 is known to hold for quite a few families of bipartite graphs, including complete bipartite graphs, even cycles, the cube graph, and etc (see [30] and [28] for some discussions). In particular, the work of Bukh and Conlon [13] on the Turán exponent of a bipartite family provides a large family of bipartite graphs H for which Conjecture 1.9 holds, namely graphs H that are obtained by gluing enough copies of a so-called balanced tree at the leaves. There is also other strong evidence that Conjecture 1.9 should be true. Bohman and Keevash [10] showed that if H is a bipartite graph 5 2− 1 that is strictly 2-balanced (see Defintion 1.6) then ex(n, H) ≥ Ω(n m2 (H) (log n)1/(e(H)−1) ). (See [28] for a slight improvement and generalization). On the other hand, a well-known conjecture of Erdős and Simonovits [19] says that for any bipartite graph H, there exist constants α ∈ [1, 2), c1 , c2 > 0 such that c1 nα ≤ ex(n, H) ≤ c2 nα . Hence, if the Erdős-Simonovits conjecture were true, then the result of Bohman and Keevash would imply Conjecture 1.9. 2 Main results As our main result of the paper, we prove Conjecture 1.4 of Morris and Saxton in a more explicit form under the same mild condition about H assumed by Ferber, McKinley and Samotij [28]. Theorem 2.1 (Balanced Supersaturation). Let r be an integer with r ≥ 2. Let H be an r-partite rgraph with h vertices and ℓ edges. Let α and A be positive reals satisfying that A ≥ r 2r , α > r− mr1(H) and that ex(n, H) ≤ Anα for all n, There exist constants k0 , C > 0 such that the following holds. Let G be an n-vertex graph with m = knα edges where k ≥ k0 . Then G contains a family H of copies of H such that , dH (S) ≤ Ck −λ(α,H)(|S|−1) where λ(α, H) = |H| , for every S ⊆ E(G), 1 ≤ |S| ≤ e(H), e(G) 1 mr (H)(r−α) . Since λ(α, H) > 1, Theorem 2.1 resolves Conjecture 1.4 in a more explicit form under the mild assumption that α > r − 1/mr (H). Our general approach for establishing Theorem 2.1 is inspired by the approach used by Ferber, McKinley and Samotij [28]. However, we also added some crucial new twists. Theorem 2.1 allows one to retrieve Theorem 1.7 and Theorem 1.8. Theorem 2.1 does not explicitly describe how dense the family H is. However, the balanced supersaturation result we will prove in fact allows one to turn any dense family of copies of H in G into a balanced one that is almost as dense. In view of that, we can obtain an even stronger version of Theorem 2.1 for those H that satisfy the following well-known conjecture of Erdős and Simonovits, which roughly says that one can expect to find asymptotically as many copies of H in G as one would expect in a random graph with the same edge-density as G. Conjecture 2.2 (Erdős-Simonovits). Let H be a bipartite graph with h vertices and ℓ edges. Let A, α be positive reals satisfying that ex(n, H) ≤ Anα for all n ∈ N. There exists constant C, c, depending on H such that for all sufficiently large n if G is an n-vertex graph with e(G) > Cnα edges then G contains at least c[e(G)]ℓ /n2ℓ−h copies of H. Given a bipartite graph H, we say that H is Erdős-Simonovits good if it satisfies Conjecture 2.2. There are quite a few known Erdős-Simonovits good graphs for appropriate values of α, for instance, even cycles [26] (see also [44]), complete bipartite graphs [24], bipartite graphs that have a vertex complete to the other part [15], tree blowups [31], tree degenerate graphs [35], etc. For ErdősSimonovits good H, we obtain the following stronger theorem. 6 Theorem 2.3 (Balanced supersaturation for Erdős-Simonovits good graphs). Let H be a bipartite graph with h vertices and ℓ edges. Suppose that H is Erdős-Simonovits good. Let α and A be positive reals satisfying that A ≥ 16, α > 2− m21(H) and that ex(n, H) ≤ Anα for all n. There exist constants δH , k0 , C > 0 such that the following holds. Let G be an n-vertex graph with m = knα edges where k ≥ k0 . Then G contains a family H of copies of H satisfying that 1. |H| ≥ δH [e(G)]ℓ /n2ℓ−h , 2. ∀S ⊆ E(G), 1 ≤ |S| ≤ E(G), dH (S) ≤ Cβ |S|−1 |H| , e(G) where ∗ (α,H) β = max{(1/k)n−φ(α,H) , k−λ φ(α, H) = αℓ−α+h−2ℓ , ℓ−1 λ∗ (α, H) = }, 1 m∗2 (H)(2−α) . Equipped with Theorem 2.1 and Theorem 2.3, we then apply them under the framework of Morris and Saxton [44] to obtain general bounds on ex(G(n, p), H) as follows. Theorem 2.4. Let H be a bipartite graph with h vertices and ℓ edges. Let A, α be positive reals such that ex(n, H) ≤ Anα for every n ∈ N. There exists a constant C = C(H) such that  n2− m21(H) ex(G(n, p), H) ≤ 1 p1− λ(α,H) nα if p ≤ n 1 2 (H) −m , otherwise with high probability as n → ∞. Theorem 2.5. Let H be a bipartite graph with h vertices and ℓ edges such that H is ErdősSimonovits good. Let A, α be positive reals such that ex(n, H) ≤ Anα for every n ∈ N. There exists a constant C = C(H) such that ex(G(n, p), H) ≤ ∗  nα−φ(α,H) (log(n))2 if p ≤ n 1 p1− λ∗ (α,H) nα (α,H) − φ(α,H)λ λ∗ (α,H)−1 2λ∗ (α,H) · (log n) λ∗ (α,H)−1 otherwise with high probability as n → ∞, where φ(α, H) = αℓ−α+h−2ℓ , ℓ−1 λ∗ (α, H) = 1 m∗2 (H)(r−α) . Theorem 2.5 implies Morris and Saxton’s result on ex(G(n, p), C2ℓ ) (see Corollary 4.11). To the best of our knowledge, Theorem 2.4 and Theorem 2.5 appear to the first general results on ex(G(n, p), H) for bipartite H. The rest of the paper is organized as follows. In Section 3, we derive our results on balanced supersaturation. In Section 4, we apply our supersaturation results to derive general bounds on the random Turán problem. 7 3 Balanced supersaturation We start with a standard estimation lemma. Lemma 3.1. Let n ≥ w ≥ h be positive integers. Then
n
n−h w ≥ h2 then w−h / w ≥ (1/2)(w/n)h . Proof. We have Hence, n−h w−h
/ n−h
w−h n
w n w
n−h
w−h n
w = n−h w−h
/ n w
≤ (w/n)h . Furthermore, if w(w − 1) . . . (w − h + 1) . n(n − 1) · · · (n − h + 1) < (w/n)h . If w ≥ h2 , then

wh − h2 wh−1 wh [1 − h2 (1/w)] 1 wh w(w − 1) · · · (w − h + 1) > > > . > h h h n n n 2 nh The following simple lemma is folklore. We include a proof for completeness. Lemma 3.2. Let r ≥ 2. Let H be an r-graph. Let G be an n-vertex r-graph with e(G) > ex(n, H). Then G contains at least e(G) − ex(n, H) different copies of H. Proof. Suppose G contains exactly m copies of H. Then we can find a set S of at most m edges whose removal destroys all the copies of H in G. Hence e(G) − m ≤ ex(n, H) and thus m ≥ e(G) − ex(n, H). We next give a nontrivial lower bound on the number of copies of any given r-partite r-graph in a dense enough r-graph. This lower bound may be of independent interest. Lemma 3.3. Let r be an integer with r ≥ 2. Let H be an r-partite r-graph with h vertices and ℓ edges. Let α and A be positive reals satisfying that A ≥ r 2r , α > r − mr1(H) and that ex(n, H) ≤ Anα for all n. There exists a constant cH > 0 such that the following holds. Let G be an n-vertex graph α(h−r) h−α with knα edges where k ≥ 23r A. Then G contains at least cH [e(G)] r−α /n r−α copies of H. Proof. Let p be a real such that 1 1 (8A/k) r−α ≤ p ≤ 2(8A/k) r−α and Such p exists since 1 1 (8A/k) r−α ≥ (8A/nr−α ) r−α ≥ 8 np ∈ Z+ . r2 , n by our condition on A. Since k ≥ 23r A, p ∈ (0, 1). Let W be a uniform random subset of V (G) of size w = np. By our choice of p, we have w ≥ r 2 . By Lemma 3.1,     1 n−r n 1 E[e(G[W ])] = e(G) / ≥ e(G)(w/n)r = e(G)pr . 2 2 w−r w (1) n
n
Let t = w and W1 , W2 , . . . , Wt be all the w subsets of V (G) of size w. For each i ∈ [t], let Gi = G[Wi ]. Let Igood be the set of i ∈ [t] such that e(Gi ) ≥ 41 mpr and Ibad = [t] \ Igood . Then P n
1 r i∈Ibad e(Gi ) ≤ w 4 e(G)p and hence by (1) X i∈Igood   n 1 e(Gi ) ≥ e(G)pr . w 4 (2) For each i ∈ Igood , we have 1 1 1 1 e(Gi ) ≥ e(G)pr = knα pr = kpr−α (np)α = kpr−α wα ≥ 2Awα , 4 4 4 4 1 r−α . where the last inequality holds since p ≥ ( 8A Hence e(Gi ) ≥ 2Awα > 2 ex(w, H). By k ) Lemma 3.2, Gi contains at least e(Gi ) − ex(n, H) ≥ 12 e(Gi ) copies of H. Let λ denote the number of copies of H in G. Then, using (2), we have X 1 1 e(Gi ) ≥ λ ≥ n−h
2 8 w−h 1 i∈Igood n
w
e(G)pr n−h w−h 1 1 1 ≥ (n/w)h e(G)pr = e(G)pr−h = e(G)(1/p)h−r . 8 8 8 1 Since k = e(G)/nα and p ≤ 2(8A/k) r−α , we have α(h−r) h−r h−α 1 λ ≥ e(G)(1/2)h−r (k/8A) r−α ≥ cH [e(G)] r−α /n r−α , 8 for some constant cH > 0. While Lemma 3.3 gives a reasonably dense family of copies of H in a dense enough host graph G, it is generally not as dense as what is conjectured in Conjecture 2.2. Next, we present our key lemma, which is the basis of our main results in this paper. Lemma 3.4 (Key Lemma). Let r, h, ℓ be positive integers, where h ≥ r ≥ 2. Let H be an r-partite r-graph with h vertices and ℓ edges. Let α, A be positive reals such that for each n every n-vertex graph G with m ≥ Anα edges contains at least f (n, m) copies of H, where f is a function satisfying the following. 1. There is a constant δ > 0 such that for all p ∈ (0, 1] and all positive reals n, m h−α f (n, m) ≥ δm r−α /n α(h−r) r−α and 2. For fixed n, f (n, m) is increasing and convex in m. 9 f (np, mpr ) ≥ δf (n, m)ph . Then there exist constants k0 = k0 (H) and C = C(H) such that if G is an n-vertex r-graph with knα edges, where k ≥ k0 , then G contains a family F of copies of H satisfying that 1. |F| ≥ δf (n, 81 e(G)). 2. ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ Cβ |S|−1 where β = max λ∗ (α, H) = 1 m∗r (H)(r−α) ( e(G) |F|  1 ℓ−1 |F| , e(G) ,k −λ∗ (α,H) ) , and m∗r (H) is the proper r-density of H. 3. Cβ ℓ−1 |F| ≥ 1. e(G) Proof. First we define some constants. Let h C = 2ℓ+h+2 r 2h (8A) r−α ℓ. (3) Let k0 be a sufficiently large constant, depending on H and A, α, such that the statement after (10) holds. Let N = δf (n, 18 e(G)). By our assumption about f and that nα = e(G)/k, α(h−r) h−α h−r 1 h−α h−r 1 N ≥ δ · δ( e(G)) r−α /n r−α = δ2 ( ) r−α k r−α e(G) = δ′ k r−α e(G), 8 8 (4) h−α where δ′ = δ2 ( 18 ) r−α . Let λ∗ (α, H) = 1 m∗r (H)(r−α)   1 e(G) ℓ−1 −λ∗ (α,H) β = max{ ,k }, N (5) and m∗r (H) is the proper r-density of H. By our choice of β, we have β ℓ−1 · N ≥ 1. e(G) (6) Let F = ∅. We show that as long as |F| < N , we can find a new copy of H in G to add to F so that the following condition holds ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ Cβ |S|−1 N . e(G) (7) Clearly, initially F satisfies (7). Given a set S ⊆ E(G), where 1 ≤ |S| ≤ ℓ − 1, we call S saturated N . Note that by (6), for all S with 1 ≤ |S| ≤ ℓ − 1, we have if dF (S) ≥ Cβ |S|−1 2e(G) Cβ |S|−1 N N C C ≥ β ℓ−1 ≥ ≥ 2. 2e(G) β 2e(G) 2β 10 (8) For each i = 1, . . . , ℓ − 1, let Bi denote the family of saturated i-subsets of E(G). We will call a Sℓ−1 Bi . copy of H in G a good copy of H if it does not contain any member of i=1 Claim 1. For each i = 1, . . . , ℓ − 1, |Bi | ≤ (2ℓ /C)(1/β)i−1 e(G). Proof of Claim 1. Let µ denote the number of of pairs (H ′ , S) where H ′ is a member of F and S ∈ Bi and S ⊆ E(H ′ ). If we count µ by S, then by definition, µ ≥ |Bi |Cβ i−1 N . 2e(G) On the other hand, if we count µ by H ′ , then   ℓ µ ≤ |F| < 2ℓ−1 N. i The claim follows by combining the last two inequalities and solving for |Bi |. 1 ∗ ∗ (α,H) Let q = (8A) r−α r 2 β mr (H) . Since β ≥ k−λ 1 ∗ (α,H)m∗ (H) r q ≥ (8A) r−α r 2 k−λ  , we have 1 1 1 = (8A) r−α r 2 k− r−α ≥ r 2 (nr−α )− r−α = r2 . n (9) Let p be a positive real such that q ≤ p ≤ 2q and np ∈ Z+ . (10) Since q ≥ r 2 /n, it is easy to see such a p exists. By (4), (5) and the fact that k ≥ k0 , by choosing k0 to be large enough constant depending on H and A, we can ensure that 2q < 1 and hence p ∈ (0, 1). Let W be a uniform random subset of V (G) of size w = np. By (9), w ≥ r 2 . By Lemma 3.1, n−r
np−r n
np E[e(G[W ])] = e(G) 1 ≥ e(G)pr . 2 (11) For each i = 1, . . . , ℓ − 1, let Yi (W ) denote the set of members of Bi that are contained in W . Fix any i = 1, . . . , ℓ − 1. Consider any member S of Bi . Suppose S spans vS vertices. Then by the ≤ m∗r (H) and hence vS ≥ r + mi−1 definition of m∗r (H), we have vi−1 ∗ (H) . Hence, S −r r P[S ⊆ W ] =     n n − vS r+ i−1 / ≤ pvS ≤ p m∗r (H) . np − vS np This, along with Claim 1, implies that for each i = 1, . . . , ℓ − 1, E[|Yi (W )| ≤ |Bi |p r+ mi−1 ∗ (H) r ≤ (2ℓ /C)(1/β)i−1 p r+ mi−1 ∗ (H) r 11 i−1 e(G) = (2ℓ /C)(1/β)i−1 p m∗r (H) · e(G)pr . (12) Hence, by (10) and (12), for each i = 1, . . . , ℓ − 1, we have 1 i−1 1 E[|Yi (W )|] ≤ (2ℓ /C)[2(8A) r−α r 2 ] m∗r (H) · e(G)pr < (2ℓ /C)[2(8A) r−α r 2 ]h · e(G)pr . By our choice of C, given in (3), we have E[| ℓ−1 [ 1 Yj (W )|] ≤ (ℓ − 1)(2ℓ /C)[2(8A) r−α r 2 ]h · e(G)pr ≤ j=1 1 e(G)pr . 4 (13) By (11) and (13), E[e(G[W ]) − | ℓ−1 [ 1 1 1 e(G)pr − e(G)pr ≥ e(G)pr , 2 4 4 Yj (W )|] ≥ j=1 (14)
n Let t = np and W1 , . . . , Wt be all the np-subsets of V (G). Let Igood be the set of i ∈ [t] such that Sℓ−1 e(G[Wi ]) − | j=1 Yj (Wi )| ≥ 18 e(G)pr . Let Ibad = [t] \ Igood . Then X [e(G[Wi ]) − | ℓ−1 [ Yj (Wi )|] ≤ j=1 i∈Ibad   n 1 e(G)pr . np 8 Hence, by (14), X e(G[Wi ]) − | ℓ−1 [ Yj (Wi )| ≥ j=1 i∈Igood   n 1 e(G)pr . np 8 (15) For each i ∈ Igood , let G′i be a subgraph of G[Wi ] obtained by deleting an edge from each member Sℓ−1 of j=1 Yj [Wi ]. By (15),   X n 1 ′ e(Gi ) ≥ e(G)pr . (16) np 8 i∈Igood By the definition of Igood , for each i ∈ Igood , we have 1 1 1 1 e(G′i ) ≥ e(G)pr = knα pr = kpr−α (np)α = kpr−α wα . 8 8 8 8 (17) By (9) and (10), 1 1 p ≥ q ≥ (8A) r−α r 2 k− r−α . Hence, by (17), for each i ∈ Igood , 1 e(G′i ) ≥ k(8A)r 2(r−α) k−1 wα ≥ Awα . 8 Hence, by our assumption about H, for each i ∈ Igood , G′i contains at least f (w, e(G′i )) = f (np, e(G′i )) copies of H in G. Now, the crucial observation is that any copy H ′ of H in G′i , where i ∈ Igood , is 12 a good copy of H in G. Indeed, suppose H ′ contains a member S of Bj for some j = 1, . . . , ℓ − 1, then S ⊆ E(G′ ) ⊆ E(G[Wi ]). So S ∈ Yj [Wi ]. But in forming G′i from G[Wi ] we have removed an edge from each member of Yj [Wi ] and hence S 6⊆ E(G′i ), a contradiction. Let Hgood denote the family of good copies of H in G. By our discussions above, (16) and our assumptions about the function f , we have |Hgood | ≥ 1 n−h
X f (np, e(G′i )) ≥ np−h i∈Igood 1 n−h
np−h   n 1 f (np, e(G)pr ) 8 np 1 1 ≥ (1/p)h δf (n, e(G))ph = δf (n, e(G)). 8 8 Hence, |Hgood | ≥ N > |F|. So, there must exist a member H ′ of Hgood that is not in F. Let us add H ′ to F. Consider any subset S ⊆ E(G), where 1 ≤ |S| ≤ ℓ − 1. If S is not contained in H ′ then dF (S) is unchanged. If S ⊆ E(H ′ ), then since H ′ is good, S is unsaturated prior to the addition of N N + 1 ≤ Cβ |S|−1 e(G) , by (8). Hence (7) still holds H ′ and hence now satisfies dF (S) ≤ Cβ |S|−1 2e(G) for the new family F. Thus, we can iterate the process to find a family F that satisfies (7) and such that |F| ≥ N . For the final F, by (6) we have Cβ ℓ−1 |F| ≥ C · 1 ≥ 1, e(G) as desired. For convenience, we will refer to the β associated with the family F as the codegree ratio of F. By Lemma 3.3 and Lemma 3.4, we obtain the following general balanced supersaturation theorem, which implies our first main theorem, Theorem 2.1. Theorem 3.5. Let r be an integer with r ≥ 2. Let H be an r-partite r-graph with h vertices and ℓ edges. Let α and A be positive reals satisfying that A ≥ r 2r , α > r − mr1(H) and that ex(n, H) ≤ Anα for all n, There exist constants δH , k0 = k0 (H), C = C(H) > 0 such that the following holds. Let G be an n-vertex graph with knα edges where k ≥ k0 . Then G contains a family F of copies of H satisfying that h−α 1. |F| ≥ δH [e(G)] r−α /n α(h−r) r−α , 2. ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ Ck −λ(α,H)(|S|−1) where λ(α, H) = 1 mr (H)(r−α) |F| , e(G) and mr (H) is the r-density of H. 3. Ck −λ(α,H)(ℓ−1) 13 |F| ≥ 1. e(G) Proof. By choosing k0 to be at least 23r , by Lemma 3.3, H has the property that every n-vertex α(h−r) h−α graph with knα edges, where k ≥ k0 contains at least cH [e(G)] r−α /n r−α copies of H. Let h−α α(h−r) f (n, m) = cH m r−α /n r−α . It is straightforward to see that f (np, mpr ) = f (n, m)ph , for any p. Furthermore, for fixed n, f (n, m) is clearly increasing and convex in m. By Lemma 3.4, there exist constants k0 and C ′ such that for every n if G is an n-vertex r-graph with knα edges, where k ≥ k0 , then G contain a family F of copies of H satisfying (a) |F| ≥ cH f (n, 81 e(G)). (b) ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ C ′ β |S|−1 where β = max λ(α, H) = ( e(G) |F|  1 ℓ−1 |F| , e(G) , k−λ(α,H) ) , 1 mr (H)(r−α) . (c) C ′ β ℓ−1 |F| ≥ 1. e(G) Since e(G) = knα , by condition (a) above, h−α |F| ≥ cH [e(G)] r−α /n α(h−r) r−α h−r = cH k r−α e(G). For convenience, we may further assume that cH < 1. By definition of r-density, Hence, by (18),  1  h−r 1 1 e(G) ℓ−1 − r−α ≤ (c−1 ) ℓ−1 ≤ (cH )− ℓ−1 k−λ(α,H) . H k |F| (18) ℓ−1 h−r ≤ mr (H). By the definition of β in condition (b) and (19), − 1 β ≤ cH ℓ−1 k−λ(α,H) . Let C = C ′ (cH )−1 . By condition (b) above, we have ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ C ′ β |S|−1 − |S|−1 |F| |F| |F| ≤ C ′ (cH ℓ−1 k−λ(H)(|S|−1)| ) ≤ Ck−λ(α,H)(|S|−1)| . e(G) e(G) e(G) By condition (c) above, we have 1 ≤ C ′ β ℓ−1 |F| |F| −1 −λ(α,H)(ℓ−1) |F| k ≤ C ′ δH = Ck −λ(α,H)(ℓ−1) . e(G) e(G) e(G) So, the theorem holds with δH = cH . 14 (19) If one applies Theorem 3.5 to an adaption of Proposition 1.5, one can retrieve the enumeration result of Ferber, McKinley and Samotij [28] as below. The difference is that Ferber, McKinley and Samotij used a weaker version of balanced supersaturation. So their constant C would be larger. Corollary 3.6. Let H be an r-uniform hypergraph and let α and A be positive constants. Suppose that α > r − mr1(H) and that ex(n, H) ≤ Anα for all n. Then there exists a constant C depending α only on α, A, and H such that for all n, there are at most 2Cn H-free r-uniform hypergraphs on n vertices. Next, we give a stronger balanced supersaturation theorem for Erdős-Simonovits good bipartite graphs. This implies our second main result, Theorem 2.3. Theorem 3.7. Let H be a bipartite graph with h vertices and ℓ edges. Suppose that H is ErdősSimonovits good. Let α and A be positive reals satisfying that A ≥ 16, α > 2 − m21(H) and that ex(n, H) ≤ Anα for all n. There exist constants δH , k0 = k0 (H), C = C(H) > 0 such that the following holds. Let G be an n-vertex graph with knα edges where k ≥ k0 . Then G contains a family F of copies of H satisfying that 1. |F| ≥ δH [e(G)]ℓ /n2ℓ−h , 2. ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ Cβ |S|−1 |F| , e(G) where ∗ (α,H) β = max{k−1 n−φ(α,H) , k−λ φ(α, H) = αℓ−α+h−2ℓ , ℓ−1 λ∗ (α, H) = 1 m∗2 (H)(2−α) 3. Cβ ℓ−1 }, and m∗2 (H) is the proper 2-density of H. |F| ≥ 1. e(G) Proof. Since H is Erdős-Simonovits good, there exist constants cH , k1 > 0 such that n-vertex graph with m = knα edges, where k ≥ k1 contains at least cH mℓ /n2ℓ−h copies of H. Let f (n, m) = cH mℓ /n2ℓ−h . It is straightforward to see that f (np, mp2 ) = f (n, m)ph , for any p. Furthermore, for fixed n, f (n, m) is clearly increasing and convex in m. By Lemma 3.4, there exist constants k0 and C ′ such that if G is an n-vertex r-graph with knα edges, where k ≥ k0 , then G contain a family F of copies of H satisfying (a) |F| ≥ cH f (n, 18 e(G)). (b) ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ Ĉ β̂ |S|−1 15 |F| , e(G) where β̂ = max λ∗ (α, H) = 1 m∗2 (H)(r−α) ( e(G) |F|  1 ℓ−1 ,k −λ∗ (α,H) ) , and m∗2 (H) is the proper 2-density of H. (c) Ĉ β̂ ℓ−1 |F| ≥ 1. e(G) Since e(G) = knα , by condition (a) above, |F| ≥ cH [e(G)]ℓ /n2ℓ−h . (20) For convenience, we may further assume that cH < 1. Hence, |F|/e(G) ≥ cH [e(G)]ℓ−1 /n2ℓ−h = cH kℓ−1 nαℓ−α+h−2ℓ = cH kℓ−1 nφ(α,H)(ℓ−1) . Hence, by the definition of β̂ given in condition (b) above, − 1 ∗ (α,H) β̂ ≤ max{cH ℓ−1 k−1 n−φ(α,H) , k−λ }. Let ∗ (α,H) β = max{k−1 n−φ(α,H) , k−λ } and C = Ĉ · c−1 H . It is straightforward to verify that condition 2 and 3 hold for these choices of β and C. So the theorem holds with δH = cH . The advantage of Theorem 3.7 over Theorem 3.5 is that for Erdős-Simonovits good H, the former produces a β value that is no larger than the former and in many cases produces a smaller β, at least for a suitable range of k. To illustrate this, we show that Theorem 3.7 immediately implies Morris and Saxton’s supersaruation result on C2ℓ (Theorem 1.1) for k in a suitable range. Corollary 3.8. For every ℓ ≥ 2, there exist constants C > 0, δ > 0 and k0 ∈ N such that for each 2 n ∈ N if G is an n-vertex graph with kn1+1/ℓ edges, where k0 ≤ k ≤ n(ℓ−1) /ℓ(2ℓ−1) , then there exists a collection F of copies of C2ℓ in G such that (a) |F| ≥ δk 2ℓ n2 , (b) ∀S ⊆ E(G), 1 ≤ |S| ≤ 2ℓ − 1, dF (S) ≤ Ck 2ℓ−|S|− |S|−1 ℓ−1 n1−1/ℓ . Proof. It is known that C2ℓ is Erdős-Simonovits good for α = 1 + 1/ℓ and some A > 0, so we can 2 apply Theorem 3.7. Let G be an n-vertex graph with knα edges, with k0 ≤ k ≤ n(ℓ−1) /ℓ(2ℓ−1) . Let F be the family of copies of H in G guaranteed by Theorem 3.7. One can check that m∗2 (C2ℓ ) = 1 16 and hence λ∗ (α, C2ℓ ) = 1 1·(2−α) φ(α, C2ℓ ) = − ℓ−1 = ℓ ℓ−1 . Also, (1 + 1/ℓ)(2ℓ) − (1 + 1/ℓ) + 2ℓ − 4ℓ ℓ−1 = . 2ℓ − 1 ℓ(2ℓ − 1) ℓ ℓ 2 So β = max{(k−1 n ℓ(2ℓ−1) , k− ℓ−1 } = k− ℓ−1 , since k ≤ n(ℓ−1) /ℓ(2ℓ−1) . Condition (a), (b) readily follow from conditions (1),(2), (3) of F guaranteed in Theorem 3.7. 2 Even though Corollary 3.8 holds only for the range k0 ≤ k ≤ n(ℓ−1) /ℓ(2ℓ−1) , whereas Theorem 1.1 holds for all k0 ≤ k ≤ n1−1/ℓ , for applications to the enumeration problem and the random Turán problem on C2ℓ , it seems to be as effective (see the proof of Theorem 4.8 and Corollary 4.11). 4 Applications to the Turán problem in random graphs In this section, we apply our balanced supersaturation results to obtain some general bounds on the Turán number of a bipartite graph H in the Erdős-Rényi random graph G(n, p). In fact, once we have Theorem 3.5 and Theorem 3.7, the corresponding random Turán results will readily follow using the framework set up by Morris and Saxton [44]. Nevertheless, for completeness, we will include all the technical details. The framework is based on the container method pioneered by Balogh, Morris, and Samotij [7] and independently by Saxton and Thomason [50]. Definition 4.1. Given an r-graph F, define the co-degree function of F δ(F, τ ) = r X 1 X 1 d(j) (v), j−1 |F| τ j=2 v∈V (F ) where d(j) (v) = max{dF (S) : v ∈ S ⊆ v(F) and |S| = j}. We need the following theorem from Morris and Saxton [44], is a quick consequence of analogous theorems of Balogh, Morris and Samotij [50] Theorem 6.2 and of Saxton and Thomason [7] Proposition 3.1. Theorem 4.2. Let r ≥ 2 be an integer. Let 0 < δ < δ0 (r) be a sufficiently small real. Let F be an r-graph with N vertices. Suppose that δ(F, τ ) ≤ δ for some τ > 0. Then there exists a collection C of subsets of V (F) and a function f : V (F)(≤τ N/δ) → C such that (a) For every independent set I in F there exists T ⊂ I with |T | ≤ τ N/δ and I ⊂ f (T ). (b) e(F[C]) ≤ (1 − δ)e(F) for every C ∈ C. Proposition 4.3. Let H be a bipartite graph with h vertices and ℓ edges. Let α, A be positive reals satisfying that for each n ∈ N every n-vertex graph G with m ≥ Anα edges contains at least f (n, m) copies of H, where f is a function satisfying the following. 17 1. There is a constant δ > 0 such that for all p ∈ (0, 1] and all positive reals n, m h−α f (n, m) ≥ δm 2−α /n α(h−2) 2−α f (np, mp2 ) ≥ δf (n, m)ph . and 2. For fixed n, f (n, m) is increasing and convex in m. Let k0 , C, F and β be as guaranteed by Lemma 3.4. There exist k0∗ ∈ N and a real ε > 0 such that the following holds for every k ≥ k0∗ and every n ∈ N. Set µ= kβ . ε Given a graph G with n vertices and knα edges, there exists a function fG that maps subgraphs of G to subgraphs of G such that for every H-free subgraph I ⊂ G, (a) There exists a subgraph T = T (I) ⊂ I with e(T ) ≤ µnα and I ⊂ fG (T ), and (b) e(fG (T (I))) ≤ (1 − ε)e(G). Proof. Note that condition 1 in the proposition still holds if we replace δ with an even smaller positive real. Hence in the our proof, we may assume δ to be sufficiently small. Let F be the family guaranteed by Lemma 3.4 with codegree ratio β. Let N = v(F) = e(G) = knα . Since we will view F as a hypergraph, we will write e(F) for |F|. Set δ2 1 = τ β and ε = δ3 . Since ∀S ⊆ E(G), 1 ≤ |S| ≤ ℓ − 1, dF (S) ≤ Cβ |S|−1 e(F)/N holds, we have   ℓ−1 ℓ−1 ℓ−1 X X 1 X 1 1 X δ2 j−1 j−1 e(F) (j) ( δ2j−2 ≤ 2Cδ2 . ) · N Cβ d (v) ≤ < C e(F) τ j−1 e(F) β N j=2 j=2 v∈V (F ) (21) j=2 Also, since ∀v ∈ V (F), d(ℓ) (v) ≤ 1 and Cβ ℓ−1 e(F)/N ≥ 1, we have X 1 1 δ2 N · ℓ−1 · ( )ℓ−1 ≤ Cδ2ℓ−2 ≤ Cδ2 , d(ℓ) (v) ≤ e(F) τ e(F) β v∈V (F ) By our discussion above, we get ℓ X 1 X 1 d(j)) (v) ≤ 2Cδ2 + Cδ2 ≤ δ. δ(F, τ ) = e(F) τ j−1 j=2 (22) v∈V (F ) By Theorem 4.2, there exist a collection C of subsets of V (F) and a function fG : V (F)(≤τ N/δ) → C such that for every H-free subgraph I ⊂ G, 18 (a’) there exists T = T (I) ⊂ I with e(T ) ≤ τ N/δ and I ⊂ fG (T ), and (b’) e(F[T (I))) ≤ (1 − δ)e(F). In condition (a’) we have e(T ) ≤ τ N/δ = (β/δ5 )e(G) = (β/ε)knα = µnα , condition (a’) is equivalent to condition (a). To complete the proof, it suffices to show that if I is an independent set in F (i.e. if I is an H-free subgraph of G) we have e(fG (T (I))) ≤ (1 − ε)e(G). Let D = fG (T (I)). By condition (b′ ), e(F[D]) ≤ (1 − δ)e(F). Hence, if we delete v(F) \ D from F we lose at least δe(F) edges of F. On the other hand, by our assumption about F, each vertex of F lies in at most Ce(F)/e(G) edges of F. Hence if we delete V (F) \ C from F, we lose at most (v(F) − |D|)Ce(F)/e(G) edges of F. Hence, we have δe(F) ≤ (v(F) − |D|)Ce(F)/e(G). Solving for |D| and using v(F) = e(G), we have |D| ≥ v(F) − (δ/C)e(G) = [1 − (δ/C)]e(G) ≥ (1 − ε)e(G). In other words, we have e(fG (T (I)) ≥ (1 − ε)e(G), as desired. Remark 4.4. When we apply Proposition 4.3, we can take β = k−λ(α,H) as in Theorem 3.5 and ∗ for Erdős-Simonovits good H, we can take β = max{k−1 n−φ(α,H) , k−λ (α,H) } as in Theorem 3.7. We need an estimation lemma from [44]. Lemma 4.5 ([44]). Let M > 0, s > 0 and 0 < δ < 1. If a1 , . . . , am are reals that satisfy s = and 1 ≤ aj ≤ (1 − δ)j M for each j ∈ [m], then s log s ≤ m X P j aj aj log aj + O(M ). j=1 In what follows, by a colored graph, we mean a graph together with a labelled partition of its edge set. Next, we prove two similar theorems, by adapting the arguments given in Section 6 of MorrisSaxton [44] to fit our general balanced supersaturation results. The former applies to all bipartite graphs that contain a cycle and the latter applies to Erdős-Simonovits good bipartite graphs that contain a cycle. Theorem 4.6. Let H be a bipartite graph with h vertices and ℓ edges that contains a cycle. Let A, α be positive reals such that ex(n, H) ≤ Anα holds for every n ∈ N. There exists a constant C such that the following holds for all sufficiently large n ∈ N and k ∈ R+ . Let I(n) denote all the 19 H-free graphs on [n] and G(n, k) the collection of all graphs on [n] with at most knα edges. There exists a collection S of colored graphs with n vertices and at most Ck 1−λ(α,H) nα edges and functions g:I→S h : S → G(n, k) and with the following properties (a) ∀s ≥ 1 the number of colored graphs in S with s edges is at most  Cnα s (1− 1 )s λ(α,H)   · exp Ck1−λ(α,H) nα . (b) ∀I ∈ I(n) g(I) ⊂ I ⊂ h(g(I)) ∪ g(I). Proof. Note that since H contains a cycle, m2 (H) ≥ 1. Let I ∈ I(n). We will apply Proposition 4.3 repeatedly (with β = k−λ(α,H) , µ = kβ/ε = (1/ε)k1−λ(α,H) ). Let G0 = Kn . For sufficiently large n, G0 clearly satisfies the condition on G in Proposition 4.3. Apply Proposition 4.3, with G0 playing the role of G to obtain the function fG0 and a subset T1 of I with T1 ⊂ I ⊂ fG0 (T1 ) where T1 and fG0 (T1 ) satisfy the additional properties described in Proposition 4.3. Now, let G1 = fG0 (T1 ) \ T1 and I1 = I ∩ G1 = I \ T1 . Apply Proposition 4.3 again, with G1 playing the role of G and I1 playing the role of I to obtain the function fG1 and a subset T2 of I1 with T2 ⊂ I1 ⊂ fG1 (T2 ). We continue like this until we arrive at a graph Gm(I) with at most knα edges. Let g(I) = T1 ∪ T2 ∪ · · · ∪ Tm(I) , where elements of Ti are colored with color i. Let S(s) = {g(I) : |g(I)| = s}. Let h(g(I)) = Gm(I) . Note that h is well-defined (see [7, 44, 50] for detailed discussion). Furthermore, as g(I) ⊂ I ⊂ h(g(I)) ∪ g(I), conditions (b) is fulfilled. It remains to show (a). We begin by partitioning S(s) into sets Sm (s) where Sm (s) = {S ∈ S(s) : the edges of S are colored with m colors}. For each m ∈ N, let K(m) = {k = (k1 , · · · km ) : kj ∈ R, (1 − ε)j−m k ≤ kj ≤ (1 − ε)j n2−α and kj nα ∈ N} And for each k ∈ K(m), let A(k) = {a = (a1 , . . . am ) : aj ∈ N, aj ≤ X 1 1−λ(α,H) kj and aj = s} ε j By definition each sequence in k ∈ K(m) corresponds to a potential sequence (G1 , · · · Gm ) where e(Gj ) = kj nα , as the edges of (1 − ε)j−m knα ≤ e(Gj ) ≤ (1 − ε)j n2 and by Proposition 4.3, we 1−λ(α,H) α have e(Tj ) ≤ 1ε kj n . Note further that our algorithm returns pairs (Gi , Ti ), such that each sequence of (T1 , · · · Tm ) is uniquely identified with a sequence (G1 , · · · Gm ). Thus it suffices to only count the choices of Ti . The sequence a ∈ A(k) corresponds to a sequence of sizes for Ti . Thus, we have that for a fixed m 20  m  X Y kj nα . aj X |Sm (s)| ≤ k∈K(m) a∈A(k) j=1 Since for each j ∈ [m] 1−λ(α,H) aj ≤ (1/ε)kj ≤ (1/ε)[(1 − ε)−j+1 k]1−λ(α,H) = (1/ε)(1 − ε)(λ(α,H)−1)(j−1) k1−λ(α,H) . So 1 1 kj ≤ (1/ε) λ(α,H)−1 n/ajλ(α,H)−1 . Hence  kj nα aj  ≤  ekj nα aj aj ≤  nα εaj  λ(α,H) ·a λ(α,H)−1 j Applying Lemma 4.5 with M = (1/ε)k1−λ(α,H) nα , 1 − δ = (1 − ε)λ(α,H)−1 , the product over j = 1, . . . , m is at most  ′ α  λ(α,H) ·s   λ(α,H)−1 Cn · exp C ′ k1−λ(α,H) nα , s for some C ′ = C ′ (H). Thus, |S(s)| ≤ ∞ X X λ(α,H)   X  C ′ nα  λ(α,H)−1 ·s · exp C ′ k1−λ(α,H) nα s m=1 k∈K(m) a∈A(k) Note that |K(m)| = 0 for values of m ≫ log n as (1 − ε)m n2 < k. For any fixed m = O(log n), |K(m)| = nO(log n) , as when picking any k, we have O(n2 ) choices for each coordinate and O(log n) coordinates to choose for. For any fixed k, similarly, |A(k)| = nO(log n) . Hence, ∞ X X |A(k)| = nO(log n) = exp(O(log2 n)). m=1 k∈K(m) Since m2 (H) ≥ 1, λ(α, H) ≤ 1/(2 − α). This imples (2 − α)(1 − λ) ≥ 1 − α. Since k ≤ n2−α , we have k1−λ(α,H) nα ≥ n ≫ log2 n. Hence, the theorem holds by choosing an appropriate C > C ′ . Remark 4.7. The proof of Theorem 4.6 can be adapted for r-partite r-graphs as well. However, as the random Turán problem in hypergraphs is not a focus of this paper, we opt to state Theorem 4.6 just for bipartite graphs. Theorem 4.8. Let H be a bipartite graph with h vertices and ℓ edges that contain a cycle. Let A, α be positive reals such that ex(n, H) ≤ Anα holds for every n ∈ N. Suppose that H is ErdősSimonovits good. There exists a constant C such that the following holds for all sufficiently large n ∈ N and k ∈ R+ with φ(α,H) 2 k ≤ n λ∗ (α,H)−1 /(log n) λ∗ (α,H)−1 . 21 Let I(n) denote all the H-free graphs on [n] and G(n, k) the collection of all graphs on [n] with at most knα edges. There exists a collection S of colored graphs with n vertices and at most ∗ Ck 1−λ (H) nα edges and functions g:I→S h : S → G(n, k) and with the following properties (a) ∀s ≥ 1 the number of colored graphs in S with s edges is at most  Cnα s (1− 1 )s λ∗ (α,H)   ∗ · exp Ck1−λ (α,H) nα . (b) ∀I ∈ I(n) g(I) ⊂ I ⊂ h(g(I)) ∪ g(I). Proof. Note that since H contains a cycle, m∗2 (H) ≥ 1. Let I ∈ I(n). As in the proof of Theorem 4.6 we will apply Proposition 4.6 repeatedly. Since H is Erdős-Simonovits good, we can use ∗ (α,H) β = max{k−1 n−φ(α,H) , k−λ ∗ (α,H) } and µ = kβ/ε = (1/ε) max{n−φ(α,H) , k1−λ }. let G0 = Kn . We apply Proposition 4.3, with G0 playing the role of G to obtain the function fG0 and a subset T1 of I with T1 ⊂ I ⊂ fG0 (T1 ) where T1 and fG0 (T1 ) satisfy the additional properties described in Proposition 4.3. Now, let G1 = fG0 (T1 ) \ T1 and I1 = I ∩ G1 = I \ T1 . Apply Proposition 4.3 again, with G1 playing the role of G and I1 playing the role of I to obtain the function fG1 and a subset T2 of I1 with T2 ⊂ I1 ⊂ fG1 (T2 ). We continue like this until we arrive at a graph Gm(I) with at most knα edges. Let g(I) = T1 ∪ T2 ∪ · · · ∪ Tm(I) , where elements of Ti are colored with color i. Let S(s) = {g(I) : |g(I)| = s}. Let h(g(I)) = Gm(I) . As in the proof of Theorem 4.6, h is well defined. Furthermore, as g(I) ⊂ I ⊂ h(g(I))∪g(I), conditions (b) is fulfilled. It remains to show (a). We begin by partitioning S(s) into sets Sm (s) where Sm (s) = {S ∈ S(s) : the edges of S are colored with m colors}. For each m ∈ N, K(m) = {k = (k1 , · · · km ) : kj ∈ R, (1 − ε)j−m k ≤ kj ≤ (1 − ε)j n2−α and kj nα ∈ N} And for each k ∈ K(m), A(k) = {a = (a1 , . . . am ) : aj ∈ N, aj ≤ X 1 1−λ∗ (α,H) −φ(α,H) max{kj ,n }nα and aj = s} ε j So each sequence in k ∈ K(m) corresponds to a potential sequence (G1 , · · · Gm ) where e(Gi ) = kj nα , as the edges of (1 − ε)j−m knα ≤ e(Gj ) ≤ (1 − ε)j n2 and by Proposition 4.3 and Remark 4.4, we 22 1−λ∗ (α,H) have e(Tj ) ≤ 1ε max{kj , n−φ(α,H) }nα . Note further that our algorithm returns pairs (Gi , Ti ), such that each sequence of (T1 , · · · Tm ) is uniquely identified with a sequence (G1 , · · · Gm ). Thus it suffices to only count the choices of Ti . The sequence a ∈ A(k) corresponds to a sequence of sizes for Ti . Thus, we have that for a fixed m |Sm (s))| ≤  m  X Y kj nα . aj X k∈K(m) a∈A(k) j=1 Now, given a k ∈ K(m) and a ∈ A(k), let us partition the product over j according to whether 1−λ∗ (α,H) kj < n−φ(α,H) (call this type 1) or not (call this type 2). Because |K(m)| = 0 for values of m ≫ log(n) as (1 − ε)m n2 < k, and some absolute constant C1 the product over type 1 j’s is at most     P ∗ (n2 ) j aj ≤ exp C1 · nα−φ(α,H) (log n)2 ≤ exp C1 · k1−λ (α,H) nα , φ(H) 2 where in the last step, we used the fact that k ≤ n λ∗ (α,H)−1 (log n) 1−λ∗ (α,H) . For each type 2 j, we 1−λ∗ (α,H) α 1−λ∗ (α,H) n . From this we get have kj ≥ n−φ(α,H) and thus aj ≤ 1ε kj 1 kj ≤ (1/ε) λ∗ (α,H)−1 n/ajλ 1 ∗ (α,H)−1 . Hence  kj nα aj  ≤  ekj nα aj aj ≤  nα εaj  ∗ λ∗ (α,H) ·a λ∗ (α,H)−1 j ∗ Applying Lemma 4.5 with M = k1−λ (α,H) nα , 1 − δ = (1 − ε)λ (α,H)−1 , the product over type 2 j’s is at most  λ∗ (α,H)    C2 nα λ∗ (α,H)−1 ·s ∗ · exp C2 k1−λ (α,H) nα , s for some C2 = C(H). Thus, letting C ′ = 2 max{C1 , C2 }, |Sm (s)| ≤ X λ∗ (α,H)   X  C ′ nα  λ∗ (α,H)−1 ·s ∗ · exp C ′ k1−λ (α,H) nα s k∈K(m) a∈A(k) And thus, |S(s)| ≤ ∞ X X λ∗ (α,H)   X  C ′ nα  λ∗ (α,H)−1 ·s ∗ · exp C ′ k1−λ (α,H) nα s m=1 k∈K(m) a∈A(k) As in the proof of Theorem 4.6, we have follows. P∞ m=1 P 23 k∈K(m) |A(k)| = nO(log n) . The theorem thus Theorem 4.9. Let H be a bipartite graph with h vertices and ℓ edges that contain a cycle. Let A, α be positive reals such that ex(n, H) ≤ Anα for every n ∈ N. There exists a constant C = C(H, α) such that  − 1 n2− m21(H) if p ≤ n m2 (H) , ex(G(n, p), H) ≤ 1 p1− λ(α,H) nα otherwise with high probability as n → ∞. Proof. Since ex(G(n, p), H) is an increasing function of e(G), it suffices to prove the claimed bound − 1 − 1 in the case p ≥ n m2 (H) . Given such a function p = p(n), define k = p λ(α,H) . Note that k ≤ n2−α . Suppose that there exists an H-free subgraph I ⊂ G(n, p) with m edges. Then by Theorem 4.6 with our choice of k there exist functions g, h such that g(I) ⊂ G(n, p) and G(n, p) has at least m − e(g(I)) edges of h(g(I)). The probability of this event is at most X S∈S knα m − e(S)  ·p m ≤ α Ck 1−λ(α,H) X n s=0  ≤ exp O(1) · (p   Cp λ(α,H)−1 λ(α,H) λ(α,H)−1 λ(α,H) nα λ(α,H) ·s λ(α,H)−1  s α  n +k    3pknα m−s 1−λ(α,H) α · exp Ck n · m−s    4pknα m/2 → 0, n ) · m 1−λ(α,H) α as n → ∞, as long as m ≫ max{p λ(α,H)−1 λ(α,H) nα , k1−λ(α,H) nα , pknα }, where we used that m − s ≥ m/2 under this assumption. One can check that these inequalities hold if λ(α,H)−1 − 1 m ≫ p λ(α,H) nα and p ≥ n m2 (H) . The theorem follows. Theorem 4.10. Let H be a bipartite graph with h vertices and ℓ edges such that H contains a cycle and is Erdős-Simonovits good. Let A, α be positive reals such that ex(n, H) ≤ Anα for every n ∈ N. There exists a constant C = C(H) such that ex(G(n, p), H) ≤  nα−φ(α,H) (log(n))2 1 p1− λ∗ (α,H) nα if p ≤ n − φ(α,H)λ∗ (α,H) λ∗ (α,H)−1 2λ∗ (α,H) · (log n) λ∗ (α,H)−1 otherwise with high probability as n → ∞. Proof. Since ex(G(n, p), H) is an increasing function of e(G), it suffices to prove the claimed bound ∗ in the case p ≥ n (α,H) − φ(H)λ λ∗ (α,H)−1 2λ∗ (α,H) · (log n) λ∗ (α,H)−1 . Let k = p φ(α,H) 1 − λ∗ (α,H) 2 . Note that by our choice of p, k ≤ n λ∗ (α,H)−1 /(log n) λ∗ (α,H)−1 . 24 Suppose that there exists an H-free subgraph I ⊂ G(n, p) with m edges. Then by Theorem 4.8 with our choice of k there exist functions g, h such that g(I) ⊂ G(n, p) and G(n, p) has at least m − e(g(I)) edges of h(g(I)). The probability of this event is at most ∗ X S∈S knα m − e(S)  ·p m ≤ (α,H) nα Ck 1−λX s=0  ≤ exp O(1) · (p   Cp λ∗ (α,H)−1 λ∗ (α,H) as n → ∞, as long as m ≫ max{p λ∗ (α,H)−1 λ∗ (α,H) nα  λ∗ (α,H) ·s λ∗ (α,H)−1  s α n +k λ∗ (α,H)−1 λ∗ (α,H)   3pknα m−s  1−λ∗ (α,H) α ·exp Ck n · m−s    4pknα m/2 n ) · → 0, m 1−λ∗ (α,H) α ∗ (α,H) nα , k1−λ nα , pknα } One can check that these inequalities hold if p≥n − φ(H)λ∗ (α,H) λ∗ (α,H)−1 · (log n) 2λ∗ (α,H) λ∗ (α,H)−1 and m ≫ p λ∗ (α,H)−1 λ∗ (α,H) nα . The theorem follows. As mentioned earlier, Theorem 4.10 implies Morris and Saxton’s result on C2ℓ . Corollary 4.11 ([44]). For every ℓ ≥ 2, there exists a constant C = C(ℓ) > 0 such that ex(G(n, p), C2ℓ ) ≤  n1+1/(2ℓ−1) (log(n))2 p1/ℓ n1+1/ℓ if p ≤ n−(ℓ−1)/(2ℓ−1) · (log n)2ℓ otherwise with high probability as n → ∞. Proof. It is known that C2ℓ is Erdős-Simonovits good with α = 1 + 1/ℓ. Apply Theorem 4.10 with ℓ−1 ℓ , φ(α, H) = ℓ(2ℓ−1) H = C2ℓ and α = 1 + 1/ℓ, noting that λ∗ (α, C2ℓ ) = ℓ−1 . 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