Bipartite-ness under smooth conditions

Bipartite-ness under smooth conditions arXiv:2109.01311v3 [math.CO] 30 Sep 2021 Tao Jiang ∗ Sean Longbrake† Jie Ma ‡ Abstract Given a family F of bipartite graphs, the Zarankiewicz number z(m, n, F ) is the maximum number of edges in an m by n bipartite graph G that does not contain any member of F as a subgraph (such G is called F -free). For 1 ≤ β < α < 2, a family F of bipartite graphs is (α, β)-smooth if for some ρ > 0 and every m ≤ n, z(m, n, F ) = ρmnα−1 + O(nβ ). Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, in [1] Allen, Keevash, Sudakov and Verstraëte proved that for any (α, β)-smooth family F , there exists k0 such that for all odd k ≥ k0 and sufficiently large n, any α−1 is bipartite. n-vertex F ∪ {Ck }-free graph with minimum degree at least ρ( 2n 5 + o(n)) In this paper, we strengthen their result by showing that for every real δ > 0, there exists k0 such that for all odd k ≥ k0 and sufficiently large n, any n-vertex F ∪ {Ck }-free graph with minimum degree at least δnα−1 is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families F consisting of the single graph Ks,t when t ≫ s. We also prove an analogous result for C2ℓ -free graphs for every ℓ ≥ 2, which complements a result of Keevash, Sudakov and Verstraëte in [20]. We will discuss the relations between our results and the conjecture of Erdős and Simonovits on compactness in the concluding remarks. 1 Introduction Given a family F of graphs, a graph G is called F-free if G does not contain any member of F as a subgraph. If F consists of a single graph F then we simply say that G is F -free. The Turán number of F, denoted by ex(n, F), is the maximum possible number of edges in an n-vertex F-free graph. As is well known, this function is well-understood when F consists only of non-bipartite graphs due to the celebrated Erdős-Stone-Simonovits theorem [10, 12] but is generally open when F contains bipartite graphs. For a family of graphs F, a closely related notion is the so-called ∗ 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. Email: [email protected]. Research supported by National Science Foundation grant DMS-1855542. ‡ School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. Email: [email protected]. Research supported by the National Key R and D Program of China 2020YFA0713100, National Natural Science Foundation of China grant 12125106, and Anhui Initiative in Quantum Information Technologies grant AHY150200. 2010 Mathematics Subject Classifications: 05C35, 05C38. Key Words: Turán numbers, cycles, bipartite graphs 1 Zarankiewicz number z(n, F), which is defined to be the maximum number of edges in an n-vertex F-free bipartite graph. More generally, we denote by z(m, n, F) the maximum number of edges in an m by n bipartite graph that is F-free. In a seminal paper [11], Erdős and Simonovits raised a number of intriguing conjectures on Turán numbers for bipartite graphs. One of them is the following (Conjecture 3 in [11]). Given a positive odd integer k, let Ck denote the family of all odd cycles of length at most k. Throughout this paper, we write f (n) ∼ g(n) for two functions f, g : N → R if limn→∞ f (n)/g(n) = 1. Conjecture 1.1 (Erdős-Simonovits [11]). Given any finite family F of graphs, there exists an odd integer k such that as n → ∞ ex(n, F ∪ Ck ) ∼ z(n, F). Erdős and Simonovits [11] verified the conjecture for F = {C4 } by showing that ex(n, {C4 , C5 }) ∼ 3 z(n, C4 ) ∼ ( n2 ) 2 . Keevash, Sudakov and Verstraëte [20] further confirmed this conjecture for Fℓ := {C4 , C6 , . . . , C2ℓ } where ℓ ∈ {2, 3, 5} in stronger forms and proved a related result for the chromatic number of Fℓ ∪ {Ck }-free graphs of minimum degree Ω(n1/ℓ ). In a subsequent paper [1], Allen, Keevash, Sudakov and Verstraëte provided a general approach to Conjecture 1.1 (using Scott’s sparse regularity lemma [26]), which works for the following families of bipartite graphs. Definition 1.2. Let α, β be reals with 2 > α > β ≥ 1. Let F be a family of bipartite graphs. If there exists some ρ > 0 such that for every m ≤ n, z(m, n, F) = ρmnα−1 + O(nβ ) holds, then we say that F is (α, β)-smooth with relative density ρ. We call a bipartite family F smooth if it is (α, β)-smooth for some α and β. It is evident to note that for any (α, β)-smooth family F, we have z(n, F) = ρ(n/2)α + O(nβ ). Before we mention the results of [1], let us discuss some known examples of smooth families. Improving results of Kövári-Sós-Turán [23], Füredi [14] showed that if m ≤ n and s, t ∈ N then z(m, n, Ks,t ) ≤ (t − s + 1)1/s mn1−1/s + sm + sn2−2/s . (1) This together with the constructions of Brown [6] and Füredi [15] shows that K2,t and K3,3 are smooth families (see [1]). Allen et al. [1] also showed that {K2,t , Bt } is smooth, where Bt consists of t copies of C4 sharing an edge (and no other vertices). However, it is not known if Ks,t is smooth for any s ≥ 3 and t ≥ 4 and if C2ℓ is smooth for any ℓ ≥ 3, due to a lack of constructions that asymptotically match upper bounds on Zarankiewicz numbers. We would like to point out that not all families of bipartite graphs are smooth – in the concluding remarks we provide an example of bipartite graphs which are not smooth. The main result of Allen et al. [1] is as follows. A family G of graphs is near-bipartite if every graph G ∈ G has a bipartite subgraph H such that e(G) ∼ e(H) as |V (G)| → ∞. Theorem 1.3 (Allen-Keevash-Sudakov-Verstraëte [1]). Let F be an (α, β)-smooth family with 2 > α > β ≥ 1. There exists k0 such that if k ≥ k0 ∈ N is odd, then the family of all extremal F ∪ {Ck }-free graphs is near-bipartite and, in particular, ex(n, F ∪ {Ck }) ∼ z(n, F). 2 Allen et al. [1] also raised a question whether the extremal n-vertex F ∪ {Ck }-free graph in Theorem 1.3 is exactly bipartite when n is sufficiently large. Motivated by the classic result of Andrásfai, Erdős and Sós [4] stating that any n-vertex triangle-free graph with minimum degree more than 2n/5 must be bipartite, Allen et al. [1] proved the following theorem, which answers their question for extremal graphs satisfying appropriate minimum degree condition. Theorem 1.4 (Allen-Keevash-Sudakov-Verstraëte [1]). Let F be an (α, β)-smooth family with relative density ρ and 2 > α > β ≥ 1. Then there exists k0 such that for any odd k ≥ k0 and sufficiently α−1 is bipartite. large n, any n-vertex F ∪ {Ck }-free graph with minimum degree at least ρ( 2n 5 + o(n)) In this paper, we strengthen Theorem 1.4 by showing that the minimum degree condition can be lowered to δnα−1 for any given real δ > 0 and furthermore, the condition on smoothness can be relaxed to the following notion. Definition 1.5. Let α, β be reals with 2 > α > β ≥ 1. Let F be a family of bipartite graphs. We say that F is (α, β)-quasi-smooth with upper density ρ and lower density ρ0 , if there exist constants ρ, ρ0 > 0 and C such that for all positive integers m ≤ n, z(m, n, F) ≤ ρmnα−1 + Cnβ and ex(n, F) ≥ ρ0 nα . If F consists of a single graph F , then we just say that F is (α, β)-quasi-smooth. The collection of (α, β)-quasi-smooth families is conceivably broader than the one of (α, β)-smooth families. For instance, it is proved that ex(n, Ks,t ) = Ω(n2−1/s ) for t ≥ (s − 1)! + 1 in [2, 21] and for t ≥ C s in a very recent paper of Bukh [7] (where C is a constant). Hence in view of (1), we know that Ks,t is quasi-smooth under these conditions. The following is our main result in this paper. Theorem 1.6. Let F be an (α, β)-quasi-smooth family with 2 > α > β ≥ 1. For any real δ > 0, there exists a positive integer k0 such that for any odd integer k ≥ k0 and sufficiently large n, any n-vertex F ∪ {Ck }-free graph with minimum degree at least δnα−1 is bipartite. The proof of Theorem 1.6 uses expansion properties and a robust reachability lemma that is in part inspired by a lemma in a recent paper by Letzter [24] on the Turán number of tight cycles. Theorem 1.6 also yields a strengthening of Theorem 1.3, which we will discuss in the concluding remarks (Theorem 5.3). We also prove an analogous result for C2ℓ -free graphs, which complements the following result in Keevash-Sudakov-Verstraëte [20]: For any integer ℓ ≥ 2, odd integer k ≥ 4ℓ + 1 and any real δ > 0, the chromatic number of any n-vertex {C4 , C6 , ..., C2ℓ , Ck }-free graph with minimum degree at least δn1/ℓ is less than (4k)ℓ+1 /δℓ . Theorem 1.7. Let ℓ ≥ 2 be an integer. For any real δ > 0, let k0 = 3ℓ(8ℓ/δ)ℓ + 2ℓ + 2. Then for any odd integer k ≥ k0 and sufficiently large n, any n-vertex {C2ℓ , Ck }-free graph with minimum degree at least δn1/ℓ is bipartite. This proof follows the same line as that of Theorem 1.6, except that we will use a more efficient robust reachability lemma for C2ℓ -free graphs and as a result get better control on k0 . 3 We should point out that the existence of such graphs in Theorem 1.7 is known only for ℓ ∈ {2, 3, 5} (see [16]). Also note that this result is not covered by Theorem 1.6, since C2ℓ is not known to be (α, β)-quasi-smooth for any ℓ ≥ 3. In the concluding remarks, we will mention that Theorems 1.6 and 1.7 can be extended to a slightly broader family of bipartite graphs that include both (α, β)quasi-smooth graphs and C2ℓ ’s. The rest of the paper is organized as follows. In Section 2, we develop some useful lemmas. In Section 3, we develop a lemma for C2ℓ -free graphs. In Section 4, we prove Theorem 1.6 and Theorem 1.7, respectively. In Section 5, we give some concluding remarks. Throughout this paper, we denote [k] by the set {1, 2, ..., k} for positive integers k. 2 Some general lemmas The main content of this section is to present key lemmas for our main result Theorem 1.6. Definition 2.1. Let α, β be reals with 2 > α > β ≥ 1. Let ℓ0 (α, β) be defined as follows:   (2 − β)(α − 1) ℓ0 = logβ + 2, for β > 1 and ℓ0 = ⌊1/(α − 1)⌋ + 1, for β = 1. α−β Lemma 2.2. Let α, β be reals with 2 > α > β ≥ 1 and ℓ0 = ℓ0 (α, β) be defined as in Definition 2.1. Let F be an (α, β)-quasi-smooth family of bipartite graphs that satisfies z(m, n, F) ≤ ρmnα−1 + Cnβ for all m ≤ n. For any δ > 0, there exists a positive real µ = µ(F, δ) such that for all sufficiently large n the following is true. Let G be an F-free bipartite graph with at most n vertices and minimum degree at least δnα−1 . Let u ∈ V (G). For each i ∈ N, let Ni (u) denote the set of vertices at distance i from u. Then for some j0 ≤ ℓ0 we have min{|Nj0 (u)|, |Nj0 +1 (u)|} ≥ µn. Proof. For each i ∈ N, let Bi denote the set of vertices at distance at most i from u. Let o n γ = (δ/12ρ)1/(α−1) and µ = min (1/2)(δ/2ρ)1/(α−1) , (δ/4ρ)γ 2−α , γ/ℓ0 . (2) First, we show that |Bℓ0 | ≥ γn. Suppose for a contradiction that |Bℓ0 | < γn. Let i ∈ [ℓ0 − 1]. Then clearly |Bi | < γn, and since G has minimum degree at least δnα−1 , we have X d(v) ≥ δnα−1 |Bi |. (3) v∈Bi P On the other hand, v∈Bi d(v) = 2e(Bi ) + e(Bi , Bi+1 \ Bi ). Since G is bipartite and F-free, e(Bi ) ≤ max(a,b) {ρabα−1 + Cbβ } over all pairs of positive integers a ≤ b with a + b = |Bi |. Hence, e(Bi ) ≤ ρ|Bi |α + C|Bi |β ≤ 2ρ|Bi |α , when n is sufficiently large. With some generosity, we can upper bound e(Bi , Bi+1 \ Bi ) by z(|Bi |, |Bi+1 |, F) to get e(Bi , Bi+1 \ Bi ) ≤ ρ|Bi ||Bi+1 |α−1 + C|Bi+1 |β . Putting the above estimations all together, we get X d(v) ≤ 4ρ|Bi |α + ρ|Bi ||Bi+1 |α−1 + C|Bi+1 |β . (4) v∈Bi 4 Combining (3) and (4), we get δnα−1 |Bi | ≤ X d(v) ≤ 4ρ|Bi |α + ρ|Bi ||Bi+1 |α−1 + C|Bi+1 |β . (5) v∈Bi If the first term on the right-hand side of (5) is the largest term, then we get δnα−1 |Bi | ≤ 12ρ|Bi |α , from which we get |Bi | ≥ (δ/12ρ)1/(α−1) n ≥ γn, contradicting our assumption. If the second term on the right-hand side of (5) is the largest term, then we get δnα−1 |Bi | ≤ 3ρ|Bi ||Bi+1 |α−1 , from which we get |Bi+1 | ≥ (δ/3ρ)1/(α−1) n ≥ γn and hence |Bℓ0 | ≥ |Bi+1 | ≥ γn, contradicting our assumption. Hence we may assume that for each i ∈ [ℓ0 − 1], we have δnα−1 |Bi | ≤ 3C|Bi+1 |β , which yields that for each i ∈ [ℓ0 − 1], |Bi+1 | ≥ (δ/3C)1/β n(α−1)/β |Bi |1/β . (6) Let {bi } be a sequence recursively defined by letting b1 = α − 1 and bi+1 = (1/β)bi + (α − 1)/β for each i ≥ 1. If β = 1 then a closed form formula for bi is bi = (α − 1)i. If β > 1 then a 1 i−1 closed form formula for bi is bi = α−1 (α − 1 − α−1 β−1 + ( β ) β−1 ). Note that we may assume C ≥ 1, so α−1 b |B1 | ≥ δn ≥ (δ/3C)n 1 . Then it follows by (6) and induction that |Bi | ≥ (δ/3C)i nbi for each i ∈ [ℓ0 ]. However, using the definition of ℓ0 we get bℓ0 > 1, which yield |Bℓ0 | > n as n is sufficiently large. This is a contradiction and thus proves that |Bℓ0 | ≥ γn. Let j ∈ [ℓ0 ] be the smallest index such that |Nj | ≥ (γ/ℓ0 )n. By the pigeonhole principle, such j exists. By (2), |Nj | ≥ µn. Let U = Nj and V = Nj−1 ∪ Nj+1 . We show that |V | ≥ 2µn, from which it follows that either |Nj−1 | ≥ µn or |Nj+1 | ≥ µn and thus the lemma holds with j0 = j or j0 = j −1. Since all the edges of G that are incident to U are between U and V , we have e(G[U, V ]) ≥ δnα−1 |U |. On the other hand, G[U, V ] is F-free. If |V | ≥ |U |, then we have e(G[U, V ]) ≤ ρ|U ||V |α−1 + C|V |β ≤ 2ρ|U ||V |α−1 , where the last inequality holds as |U | = |Nj | ≥ µn and n is sufficiently large. Combining the two inequalities and solving for |V |, we get |V | ≥ (δ/2ρ)1/(α−1) n ≥ 2µn, as desired. Otherwise, we have |V | ≤ |U |. Then δnα−1 |U | ≤ e(G[U, V ]) ≤ ρ|V ||U |α−1 + C|U |β . Since C|U |β ≪ δnα−1 |U | for sufficiently large n, we can derive from the above that δnα−1 |U | ≤ 2ρ|V ||U |α−1 . Solving for |V |, we have |V | ≥ (δ/2ρ)nα−1 |U |2−α ≥ (δ/2ρ)nα−1 (γn)2−α = (δ/2ρ)γ 2−α n ≥ 2µn, where the last inequality holds by (2), as desired. The following lemma, which we call robust reachability lemma is key to our proof of the main results. It is inspired by a lemma used in a recent paper of Letzter [24] on the Turán number of tight cycles in hypergraphs. Lemma 2.3. Let F be an (α, β)-quasi-smooth family of bipartite graphs with 2 > α > β ≥ 1. Let ℓ0 = ℓ0 (α, β) be defined as in Definition 2.1. For any real δ > 0, the following holds for all sufficiently large n. Let G be an F-free bipartite graph with at most n verticers and minimum degree at least δnα−1 . Let u ∈ V (G). Then there exists a set S of at least µ(F, δ/2)n vertices, and a family P = {Pv : v ∈ S}, where for each v ∈ S, Pv is a u, v-path of length at most ℓ0 , such that no vertex except u is used on more than logn n of the paths in P. 5 Proof. Let S be a maximum set of vertices such that there is an associated family P = {Pv : v ∈ S}, where for each v ∈ S, Pv is a path of length at most ℓ0 such that no vertex is on more than logn n of the paths. Let W denote the set of vertices in G (other than u) that lie on exactly logn n of the paths Pv in P. Then |W | logn n ≤ |S|ℓ0 ≤ nℓ0 and thus |W | ≤ ℓ0 log n < (δ/2)nα−1 for sufficiently large n. Hence, G − W has minimum degree at least (δ/2)nα−1 . If |S| < µ(F, δ/2)n, then by Lemma 2.2, there exists a vertex z ∈ / S and a u, z-path Pz of length at most ℓ0 in G − W that we can add to S to contradict our choice of S. Hence |S| ≥ µ(F, δ/2)n. The next folklore lemma will be used a few times and we include a proof for completeness. We would like to mention that it might be easy for one to overlook the connectedness of H statement in the conclusion. But this condition will play important role in the main proofs. Lemma 2.4. Let G be a connected graph. Let H be a maximum spanning bipartite subgraph of G. Then H is connected and for each v ∈ V (G), dH (v) ≥ (1/2)dG (v). Proof. Let (X, Y ) denote a bipartition of H. Suppose for contradiction that H is disconnected and F is a component of H. Since G is connected, it contains an edge e joining V (F ) to V (G) \ V (F ). But then H ∪ e is still bipartite, since adding e does not create a new cycle. Furthermore, H ∪ e has more edges than H, contradicting our choice of H. Next, let v be any vertex in H. Without loss of generality, suppose v ∈ X. Suppose dH (v) < (1/2)dG (v). Then from H by deleting the edges incident to x and adding the edges in G from v to X, we obtained a bipartite subgraph of G that has more edges than H, a contradiction. Hence ∀v ∈ V (G), dH (v) ≥ (1/2)dG (v). We conclude this section with the following lemma about the diameter. The diameter of a graph G is the least integer k such that there exists a path of length at most k between any two vertices in G. Lemma 2.5. Let G be an n-vertex connected graph with minimum degree at least D. Then G has diameter at most 3n/D. Proof. Let x, y be two vertices at maximum distance in G. Let v0 v1 · · · vℓ be a shortest x, y-path in G where v0 = x and vℓ = y. Let q = ⌊ℓ/3⌋. Note that N (v0 ), N (v3 ), N (v6 ), · · · , N (v3q ) are pairwise Pq disjoint (or else we can find a shorter x, y-path, a contradiction). Hence n ≥ i=0 |N (v3i )| ≥ (q + 1)D. This implies that (q + 1) ≤ n/D and hence ℓ ≤ 3(q + 1) ≤ 3n/D. 3 An efficient robust reachability lemma for C2ℓ-free graphs In this section, we develop a more efficient robust reachability lemma than Lemma 2.3 for C2ℓ -free graphs, which may be of independent interest. We need the following lemma from [27]. Lemma 3.1 (Verstraëte [27]). Let ℓ ≥ 2 be an integer and H a bipartite graph of average degree at least 4ℓ and girth g. Then there exist cycles of at least (g/2 − 1)ℓ ≥ ℓ consecutive even lengths in H. Moreover, the shortest of these cycles has length at most twice the radius of H. 6 Our lemma is as follows. Lemma 3.2. Let ℓ ≥ 2 and d be positive integers. Let H be a bipartite C2ℓ -free graph with minimum degree at least d. Let u be any vertex in H. Then the following items hold. (1). The number of vertices that are at distance at most ℓ from u is at least (d/4ℓ)ℓ . (2). Suppose H has at most n vertices and d ≥ 15ℓ log n, where n is sufficiently large. Then there is a set S of at least (1/2)(d/8ℓ2 )ℓ vertices together with a family P = {Pv : v ∈ S}, where for each v ∈ S, Pv is a u, v-path of exactly length ℓ, such that no vertex of H except u lies on more than dℓ−1 of these paths and each vertex v in S lies only on Pv . Proof. First we prove the first part (1) of the theorem. Let B0 = {u}. Consider any i ∈ [ℓ]. Let Bi denote the set of vertices at distance at most i from u in H and Hi the subgraph of H induced by Bi . If H[Bi ] has average degree at least 4ℓ, then by Lemma 3.1, Gi contains cycles of ℓ consecutive even lengths the shortest of which has length at most 2i ≤ 2ℓ and hence it contains C2ℓ , contradicting G being C2ℓ -free. So for each i ∈ [ℓ], we have d(Hi ) < 4ℓ, which implies that e(Hi ) < 2ℓ|Bi |. On the other hand, Hi contains all the edges of G that are incident to Bi−1 . So e(Hi ) ≥ d|Bi−1 |/2. Combining these two inequalities, we get 2ℓ|Bi | > d|Bi−1 |/2. Hence, |Bi | > (d/4ℓ)|Bi−1 | for each i ∈ [ℓ]. Thus, |Bℓ | ≥ (d/4ℓ)ℓ , as desired. Next, we prove the second part (2). Let us randomly split the vertices of G into ℓ parts V1 , . . . , Vℓ . For each vertex x, and each i ∈ [ℓ], the degree di (x) of x in Vi has a binomial distribution Bin(d(x), 1/ℓ). Hence, using Chernoff’s inequality (see [3] or [17] Corollary 2.3), we have P[di (x) < (1/2ℓ)d(x)] ≤ P[|di (x) − (1/ℓ)d(x)| > (1/2ℓ)d(x)] ≤ 2e−d(x)/12ℓ = o(n−1 ), since d(x) ≥ 15ℓ log n. Hence, for sufficiently large n, there exists a splitting of V (H) such that for each x ∈ V (H) and for each i ∈ [ℓ], di (x) ≥ (1/2ℓ)d(x) ≥ d/2ℓ. Now, we form a subgraph H ′ of H as follows. First, we include exactly d/2ℓ of the edges from u to V1 . Denote the set of reached vertices in V1 by S1 . Then for each vertex in S1 including exactly d/2ℓ edges from it to V2 . Denote the set of reached vertices in V2 by S2 . We continue like this till we define Sℓ . Let B0 = S0 = {u}. S For each i ∈ [ℓ], let Bi = ij=0 Si . and Hi the subgraph of H ′ induced by Bi . Note that Hi has radius i. As in the proof of the first part of the lemma, since Hi is C2ℓ -free, e(Hi ) < 2ℓ|Bi |. On the other hand, Hi contains all the edges of H ′ that are incident to Bi−1 . So e(Hi ) ≥ (1/2)(d/2ℓ)|Bi−1 |. Combining these two inequalities, we get 2ℓ|Bi | > (d/4ℓ)|Bi−1 |. Hence, |Bi | > (d/8ℓ2 )|Bi−1 | for each i ∈ [ℓ]. Thus, |Bℓ | ≥ (d/8ℓ2 )ℓ . Pℓ−1 Pℓ−1 It is easy to see that i=0 |Si | ≤ i=0 (d/2ℓ)i ≤ 2(d/2ℓ)ℓ−1 , when n is sufficiently large. Hence ℓ−1 |Sℓ | = |Bℓ \ ∪i=0 Si | ≥ (d/8ℓ2 )ℓ − 2(d/2ℓ)ℓ−1 > (1/2)(d/8ℓ2 )ℓ , where n (and thus d) is sufficiently large. By the definition of H ′ , for each v ∈ Sℓ , there is a path of length ℓ from u to v that intersects each of V1 , V2 , . . . , Vℓ . From the union of these paths one can find a tree T of height ℓ rooted at u, in which all the vertices in Sℓ are at distance ℓ from u. Furthermore, by the definition of H ′ , T has maximum degree at most (d/2ℓ) + 1. For each v ∈ Sℓ , let Pv be the unique u, v-path in T . If x is any vertex in T other than u, then clearly x lies on at most (d/2ℓ)ℓ−1 of the paths Pv . Furthermore, each v ∈ Sℓ doesn’t lie on any Pw for w ∈ Sℓ \{v}. 7 4 Proof of Theorem 1.6 and Theorem 1.7 Even though the proofs of Theorem 1.6 and Theorem 1.7 are essentially the same, there are sufficiently different choices of parameters that we will prove them separately. Proof Theorem 1.6: Let F be an (α, β)-quasi-smooth family with 2 > α > β ≥ 1. Given any real δ > 0, we first define k0 as following. Let ℓ0 = ℓ0 (α, β) and µ(F, δ/2) be defined as in Definition 2.1 and Lemma 2.2, respectively. Define   3 L := · ℓ0 and k0 := 2ℓ0 + L + 2. (7) µ(F, δ/2) Let k ≥ k0 be odd. Let n be sufficiently large so that all subsequent inequalities involving n hold. Let G be an n-vertex F ∪ {Ck }-free graph with minimum degree at least δnα−1 . We may assume that G is connected. Let H be a maximum bipartite spanning subgraph of G. By Lemma 2.4, H is connected and has minimum degree at least (δ/2)nα−1 . Since H is F-free, by Lemma 2.2, for each vertex x, the set of vertices that are at distance at most ℓ0 from x is at least µ(F, δ/2)n. Hence by Lemma 2.5 (applied to the ℓ0 -th power H ℓ0 of H), H ℓ0 3 3 has diameter at most ⌊ µ(F3n ,δ/2)n ⌋ = ⌊ µ(F ,δ/2) ⌋ and hence H has diameter at most ⌊ µ(F ,δ/2) ⌋ · ℓ0 = L, as defined in (7). Let (X, Y ) be the unique bipartition of H. We show that G is also bipartite with (X, Y ) being a bipartition of it. Suppose otherwise. We may assume, without loss of generality, that there exist two vertices u, v ∈ X such that uv ∈ E(G). We will derive a contradiction by finding a copy of Ck in G that contains uv. Let us randomly split V (H) into two subsets A, B. For each vertex x of degree d(x) in H, let dA (x) and dB (x) denote the degree of x in A and B, respectively. Then both dA (x) and dB (x) satisfy the binomial distribution Bin(d(x), 1/2). Hence, by Chernoff’s inequality, we have P(dA (x) < (1/4)d(x)) ≤ P(|dA (x) − d(x)/2| ≥ d(x)/4) ≤ 2e−d(x)/24 = o(n−1 ), since d(x) ≥ (δ/2)nα−1 and n is sufficiently large. Hence with positive probability we can ensure that for any x ∈ V (H), min{dA (x), dB (x)} ≥ (1/4)d(x) ≥ (δ/8)nα−1 . Let us fix such a partition A, B of V (H). Without loss of generality, suppose that the vertex u is in A. Let H[A] denote the subgraph of H induced by A. By our discussion above, H[A] has minimum degree at least (δ/8)nα−1 . By Lemma 2.3, there exists a set U of at least µ(F, δ/16)n vertices and a family P = {Pz : z ∈ U }, where for each z ∈ U , Pz is a u, z-path in H[A] of length at most ℓ0 . By the pigeonhole principle, there exists a value p ∈ [ℓ0 ] and a subset S ⊆ U of size |S| ≥ |U |/ℓ0 ≥ (µ(F, δ/16)/ℓ0 )n such that for each z ∈ S, Pz has length p and no vertex in H lies on more than n/ log n of these paths Pz . Now, let us randomly color the vertices in H[A] with colors 1 and 2. For each z ∈ S, the path Pz is good if z is colored 2 and the other p vertices on Pz are colored 1. The probability that Pz is good is 1/2p+1 . Hence, for some coloring, there are at least |S|/2p+1 good paths. Let S ′ denote 8 the subset of vertices z ∈ S such that Pz is good and let P ′ = {Pz : z ∈ S ′ }. By our discussion, |P ′ | = |S ′ | ≥ µ(F, δ/16)/(2ℓ0 +1 ℓ0 ). Note that by our definition of P ′ , no vertex in S ′ is used as an internal vertex of any path in P ′ . Let H ′ denote the bipartite subgraph of H induced by the two parts S ′ and B. By our earlier discussion, each vertex in S ′ has at least (δ/8)nα−1 neighbors in B. Hence,   e(H ′ ) ≥ |S ′ |(δ/8)nα−1 ≥ µ(F, δ/16)δ/(2ℓ0 +4 ℓ0 ) nα = γnα , where γ := γ(F, δ) = µ(F, δ/16)δ/(2ℓ0 +4 ℓ0 ). So H ′ has average degree at least 2γnα−1 . By a wellknown fact, H ′ contains a subgraph H ′′ of minimum degree at least γnα−1 . Let S ′′ = V (H ′′ ) ∩ S ′ . Let Q be a shortest path in H from the vertex v to V (H ′′ ). Let y be the endpoint of Q in V (H ′′ ). By our choice of Q, y is the only vertex in V (Q) ∩ V (H ′′ ) (note that it is possible that y = v). Let q denote the length of Q. Since H has diameter at most L, we have q ≤ L. By Lemma 2.2, for some j0 ≤ ℓ0 , inside the graph H ′′ we have min{|Nj0 (y)|, |Nj0 +1 (y)|} ≥ µ(F, γ)n. Note that one of Nj0 (y) and Nj0 +1 (y) lies completely inside V (H ′′ ) ∩ A. Denote this set by W . Let W0 = {w ∈ W : Pw ∩ V (Q) 6= ∅}. Since Q contains at most L + 1 vertices, each of which lies on at most n/ log n of the members of P, we see |W0 | ≤ (L + 1)(n/ log n). Since n is sufficiently large, we have |W \ W0 | ≥ µ(F, γ)n − (L + 1)n/ log n ≥ (1/2)µ(F, γ)n. Let H[W \ W0 , B] denote the subgraph of H consisting of edges that have one endpoint in W \ W0 and the other endpoint in B. Since each vertex in W \ W0 has at least (δ/8)nα−1 neighbors in B, e(H[W \ W0 , B]) ≥ |W \ W0 |(δ/8)nα−1 ≥ (1/16)µ(F, γ)δnα . Hence H[W \W0 , B] contains a subgraph H ∗ with minimum degree at least (1/16)µ(F, γ)δnα−1 ≥ k, for sufficiently large n. Let w be any vertex in V (H ∗ ) ∩ (W \ W0 ). By our definition of W , there is a path Rw in H ′′ of length r ≤ j0 + 1 ≤ ℓ0 + 1 from y to w. Let t = k − 1 − q − r − p. Since q ≤ L, p ≤ ℓ0 , r ≤ ℓ0 + 1 and k ≥ k0 = 2ℓ0 + L + 2, we get t ≥ 0. Since there is a path in H of length p from w to u (by the definition of S ′′ ⊆ W ) and Rw ∪ Q is a path in H of length q + r from w to v and u, v ∈ X, p and q + r have the same parity. Since k is odd, we see that t is even. Since H ∗ has minimum (much) larger than k, greedily we can build a path T of length t in H ∗ from w to some vertex w∗ in V (H ∗ ) ∩ (W \ W0 ) such that T intersects Q ∪ Rw only in w. Now, let C := uv ∪ Q ∪ Rw ∪ T ∪ Pw∗ By our definitions, Q ∪ Rw ∪ T is a path. Also, since w∗ ∈ W \ W0 , Pw∗ is vertex disjoint from Q. Finally, by our definition of P ′ , V (Pw∗ ) \ {w∗ } is disjoint from S ′ and hence from S ′′ . It is certainly also disjoint from B and hence is vertex disjoint from Rw ∪ T . So, C is a cycle in G of length 1 + q + r + t + p = k, a contradiction. 9 Proof of Theorem 1.7: Let ℓ ≥ 2 be an integer. Let δ > 0 be a real. Define L := 3ℓ(8ℓ/δ)ℓ and k0 := 2ℓ + L + 2. (8) Let k ≥ k0 be an odd integer. Let n be sufficiently large so that all subsequent inequalities involving n hold. Let G be an n-vertex C2ℓ -free graph with minimum degree at least δn1/ℓ . We may assume that G is connected. Let H be a maximum spanning subgraph of G. By Lemma 2.4, H is connected with minimum degree at least (δ/2)n1/ℓ . Since H is C2ℓ -free, by Lemma 3.2, the ℓ-th power H ℓ of H has minimum degree at least (δn1/ℓ /8ℓ)ℓ = (δ/8ℓ)ℓ n. Hence, by Lemma 2.5, H ℓ has diameter at most 3n/[(δ/8ℓ)ℓ n] = 3(8ℓ/δ)ℓ . Therefore, H has diameter at most 3ℓ(8ℓ/δ)ℓ = L, as defined in (8). Let (X, Y ) the unique bipartition of H. We show that G is also bipartite with (X, Y ) being a bipartition of it. Suppose otherwise. Then without loss of generality, we may assume that there exist two vertices u, v ∈ X such that uv ∈ E(G). We will derive a contradiction by finding a copy of Ck in G that contains uv. As in the proof of Theorem 1.6, we can split V (H) into two subsets A, B such that for each vertex x ∈ V (H), we have dA (x), dB (x) ≥ (1/4)d(x) ≥ (δ/8)n1/ℓ . Without loss of generality, suppose u ∈ A. Let H[A] denote the subgraph of H induced by A. By our discussion above, H[A] has minimum degree at least (δ/8)n1/ℓ . By Lemma 3.2 (with d = (δ/8)n1/ℓ ), there exists a set S of size at least (δ/64ℓ2 )ℓ (n/2) and a family P = {Pz : z ∈ S}, where for each z ∈ S, Pz is a u, z-path in H[A] of length ℓ, such that no vertex other than u in H lies on more than (δn1/ℓ /8)ℓ−1 = (δ/8)ℓ−1 n1−1/ℓ of these paths. Furthermore, for each z ∈ S, z lies only on Pz . Let H[S, B] denote the bipartite subgraph of H induced by the two parts S and B. By our earlier discussion, each vertex in S has at least (δ/8)n1/ℓ neighbors in B. Hence, e(H) ≥ |S|(δ/8)n1/ℓ ≥ (δℓ+1 /26ℓ+4 ℓ2ℓ )n1+1/ℓ = γn1+1ℓ , where γ := δℓ+1 /26ℓ+4 ℓ2ℓ . Then H[S, B] has average degree at least 2γn1/ℓ and thus contains a subgraph H ′ of minimum degree at least γn1/ℓ . Let S ′ = V (H ′ ) ∩ S and B ′ = V (H ′ ) ∩ B. If ℓ is even, then S ′ ⊆ X and let Q be a shortest path in H from v to S ′ . If ℓ is odd, then B ′ ⊆ X and let Q be a shortest path in H from v to B ′ . Let y denote the endpoint of Q opposing v (it is possible that y = v). Let q denote the length of Q. So q is even and y ∈ X. In either case, it is easy to see that q ≤ L + 1 and that V (H ′ ) contains y and at most one other vertex of Q. Hence H ′ − (V (Q) \ {y}) has minimum degree at least γn1/ℓ − 1 ≥ (γ/2)n1/ℓ . By Lemma 3.2 (with d = (γ/2)n1/ℓ ), inside the graph H ′ − (V (Q) \ {y}) there is a set W of size at least (1/2)(γ/16ℓ2 )ℓ n such that for each w ∈ W there is a path Rw of length ℓ from y to w in H ′ − (V (Q) \ {y}). Furthermore, by our definition of Q, we can get W ⊆ S ′ . Recall the paths Pw in P. Let W0 = {w ∈ W : Pw ∩ V (Q) 6= ∅}. Since Q contains at most L + 1 vertices each of which lies on at most (δ/8)ℓ−1 n1−1/ℓ of the members of P, we see |W0 | ≤ (L + 1)(δ/8)ℓ−1 n1−1/ℓ . Since n is sufficiently large, we have |W \ W0 | ≥ (1/2)(γ/16ℓ2 )ℓ n − (L + 1)(δ/8)ℓ−1 n1−1/ℓ ≥ (1/4)(γ/16ℓ2 )ℓ n. 10 Since each vertex in W \ W0 has at least (δ/8)nℓ neighbors in B, e(H[W \ W0 , B]) ≥ |W \ W0 |(δ/8)n1/ℓ ≥ (δ · γ ℓ )/(24ℓ+5 · ℓ2ℓ )n1+1/ℓ . Hence H[W \W0 , B] contains a subgraph H ∗ with minimum degree at least (δ ·γ ℓ )/(24ℓ+5 ·ℓ2ℓ )n1/ℓ ≥ k, for sufficiently large n. Let w be any vertex in V (H ∗ ) ∩ (W \ W0 ). By our definition of W , there is a path Rw in H ′ − (V (Q) \ {y}) of length ℓ from y to w. Let t = k − 1 − q − 2ℓ. Since q ≤ L + 1 and k ≥ k0 = 2ℓ + L + 2, we see t ≥ 0. Since k is odd and q is even, we also see that t is even. Since H ∗ has minimum degree (much) larger than k, greedily we can build a path T of length t in H ∗ from w to some vertex w∗ in V (H ∗ ) ∩ (W \ W0 ) such that T intersects Q ∪ Rw only in w. Now, let C := uv ∪ Q ∪ Rw ∪ T ∪ Pw∗ By our definition of T , Q ∪ Rw ∪ T is path. Also, since w∗ ∈ W \ W0 , Pw∗ is vertex disjoint from Q. Finally Pw∗ \ {w∗ } does not contain any vertex of S ′ ∪ B and hence is vertex disjoint from Rw ∪ T . Hence, C is a cycle in G of length 1 + q + 2ℓ + t = k, a contradiction. 5 Concluding Remarks 1. Given integers t, ℓ ≥ 2, the theta graph θt,ℓ is the graph consisting of t internally disjoint paths of length ℓ between two vertices. In particular, we have θ2,ℓ = C2ℓ . Faudree and Simonovits [13] showed that for all t, ℓ ≥ 2, ex(n, θt,ℓ ) = O(n1+1/ℓ ) (the case t = 2 was first proved by Bondy-Simonovits [5]). Conlon [9] showed that for each ℓ ≥ 2, there exists a t0 such that for all t ≥ t0 , ex(n, θt,ℓ ) = Ω(n1+1/ℓ ), the leading coefficients of which were further improved by Bukh-Tait [8]. Jiang, Ma and Yepremyan [19] showed that for all t, ℓ, there exists a constant c = c(t, ℓ) such that for all m ≤ n ( ℓ+1 c · [(mn) 2ℓ + m + n] if ℓ is odd, z(m, n, θt,ℓ ) ≤ 1 1 1 c · [m 2 + ℓ n 2 + m + n] if ℓ is even. For the case t = 2, the above bound was first proved by Naor-Verstraëte [25], and a different form of the upper bound on z(m, n, θ2,ℓ ) was obtained by Jiang-Ma [18]. On the other hand, using first moment deletion method it is not hard to show that Proposition 5.1. Let ε > 0 be any real. Let ℓ ≥ 2. There exists a t0 such that for all t ≥ t0 , if ℓ is odd then ℓ+1 ℓ+1 z(m, n, θt,ℓ ) ≥ Ω(m 2ℓ −ε n 2ℓ −ε ), and if ℓ is even then 1 1 1 z(m, n, θt,ℓ ) ≥ Ω(m 2 + ℓ −ε n 2 −ε ). Proof. Consider the bipartite random graph G ∈ G(m, n, p) with p to be chosen later. Let q = ⌊ℓ/2⌋. Let X = e(G) and Y denote the number of copies of θt,k in G. We have E(X) = mnp. If ℓ is odd, then ℓ = 2q + 1 and E[Y ] ≤ [m]tq+1 [n]tq+1 pt(2q+1) < (1/2)mtq+1 ntq+1 pt(2q+1) . −tq −tq We now choose p so that E[X] ≥ 2E[Y ]. It suffices to set p = m 2tq+t−1 n 2tq+t−1 . Since 11 E(X −Y ) ≥ (1/2)E[X], there exists a (m, n)-bipartite graph G for which X −Y ≥ (1/2)mnp = tq+t−1 (1/2)(mn) 2tq+t−1 . By deleting one edge from copy of θt,ℓ in G, we obtained a (m, n)-bipartite graph G′ that is θt,ℓ -free and satisfies tq+t−1 e(G′ ) ≥ (1/2)(mn) 2tq+t−1 = (1/2)(mn) For sufficiently large t, we have e(G′ ) ≥ (1/2)(mn) ℓ+1 −ε 2ℓ ℓ+1−(2/t) 2ℓ−(2/t) . , as desired. For even integers ℓ = 2q, the analysis is similar, except that we use the bound E[Y ] ≤ [m]tq+1 [n]t(q−1)+1 +[m]t(q−1)+1 [n]tq+1 )p2tq < (1/2)mt(q−1)+1 ntq+1 p2tq . We omit the details. It is quite likely using the random algebraic method used in [9], one could show that for each ℓ+1 ℓ+1 ℓ ≥ 2, there exist a t0 such that for all t ≥ t0 if ℓ is odd then z(m, n, θt,ℓ ) ≥ Ω(m 2ℓ n 2ℓ ) and 1 1 1 if ℓ is even then z(m, n, θt,ℓ ) ≥ Ω(m 2 + ℓ n 2 ). In any case, Proposition 5.1 already shows that θt,ℓ is not (α, β)-quasi-smooth and hence is also not (α, β)-smooth. (As far as we know, this is the first example of a family of bipartite graphs which are not (α, β)-smooth.) However, θt,ℓ -free graphs have similar expansion properties as C2ℓ -free graphs (see [19], Lemma 4.1). By using Lemma 4.1 in [19] instead of Lemma 3.1 in this paper, one can develop an analogous lemma as Lemma 3.2. Then using essentially the same proof as that of Theorem 1.7, one can show the following. Theorem 5.2. Let t, ℓ ≥ 2. Let δ > 0 be any real. Let k0 = 3ℓ(8ℓ/δ)ℓ + 2ℓ + 2. For all odd integers k ≥ k0 and n sufficiently large the following is true. If G is an n-vertex {θt,ℓ , Ck }-free graph with minimum degree at least δn1/ℓ , then G is bipartite. 2. Our proof method works for a slightly broader family than (α, β)-quasi-smooth graphs and theta graphs. Suppose F is a family of bipartite graphs satisfying the following property (P1). (P1). For any δ > 0, there are constants K and µ such that for every n-vertex F-free graph G with minimum degree δ ex(n, F)/n and for each vertex u in G, there are at least µn vertices within distance K from u. Then analogous theorems as Theorems 1.6 and 1.7 hold for F-free graphs. 3. Let F be an (α, β)-smooth family with 2 > α > β ≥ 1 and let k ≥ k0 be a (large) odd integer. Allen-Keevash-Sudakov-Verstraëte [1] proved that for any ǫ > 0 and sufficiently large n, any n-vertex extremal F ∪ {Ck }-free graph can be made bipartite by deleting at most ǫnα edges. This implies that ex(n, F ∪ {Ck }) = z(n, F) + o(nα ) and thus Theorem 1.3 holds. As a direct application of Theorem 1.6, one can strengthen Theorem 1.3 by deleting only O(n1+β−α ) vertices (of low degree) to make the extremal graph bipartite. This gives further evidence to an affirmative answer to the question of Allen et al. [1] that whether the extremal n-vertex F ∪ {Ck }-free graph G in Theorem 1.3 is bipartite (for sufficiently large n). Theorem 5.3. Let F be an (α, β)-smooth family with 2 > α > β ≥ 1. Then there exists k0 such that for any odd k ≥ k0 and sufficiently large n, any n-vertex F ∪{Ck }-free extremal graph can be made bipartite by deleting a set of O(n1+β−α ) vertices, which together are incident to O(nβ ) edges. Therefore, ex(n, F ∪ {Ck }) = z(n, F) + O(nβ ). 12 Proof. Let 2 > α > β ≥ 1 and F be an (α, β)-smooth family with relative density ρ. Then by the remark after Definition 1.2, there exist constants C1 < C2 such that for sufficiently large n, ρ(n/2)α + C1 nβ ≤ z(n, F) ≤ ρ(n/2)α + C2 nβ . Fix δ := ρ/2α+3 . Let k0 be from Theorem 1.6 such that for any odd k ≥ k0 and sufficiently large m, any m-vertex F ∪ {Ck }-free graph with minimum degree at least δmα−1 is bipartite. Now consider any odd k ≥ k0 and sufficiently large n. Let G be an n-vertex extremal F ∪{Ck }free graph. Then e(G) = ex(n, F ∪ {Ck }) ≥ z(n, F) ≥ ρ α n + C1 n β . 2α (9) Let G0 = G. If there exists some vertex x of degree less than δnα−1 in G0 , then we delete the vertex x and rename the remaining subgraph as G0 . We repeat the above process until there is no such vertex in G0 . Let H denote the remaining induced subgraph of G and let ρ t = n − |V (H)|. We note that as α < 2 and n is sufficiently large, using (9) and δ = 2α+3 , e(H) ≥ e(G) − t · δnα−1 ≥  ρ 7 ρ α α β − n · δnα−1 = n + C n n + C1 n β . 1 α 2 8 2α (10) Let m = |V (H)|. By definition, H is an m-vertex F ∪ {Ck }-free graph with minimum degree at least δnα−1 ≥ δmα−1 . By taking n sufficiently large, we can make m large enough to apply Theorem 1.6 to conclude that H is bipartite. Since H is also, F-free, we have e(H) ≤ z(m, F) = z(n − t, F) ≤ ρ (n − t)α + C2 nβ . 2α (11) Comparing (10) and (11), we see that n − t ≥ 43 n. By (9), (10) and (11), we have  ρ ρ α β − t · δnα−1 ≤ α (n − t)α + C2 nβ n + C n 1 α 2 2 Recall that δ = ρ/2α+3 . Rearranging the above inequality, we get ρ α ρ n − α (n − t)α − t · δnα−1 ≤ (C2 − C1 )nβ . α 2 2 By the Mean Value Theorem, for some n − t ≤ n′ ≤ n, this is equivalent to ρα ′ α−1 t(n ) − t · δnα−1 ≤ (C2 − C1 )nβ . 2α Since n − t ≥ 34 n and ρα 3 α−1 2α ( 4 ) >2· ρ 2α+3 , ρ 2α+3 the above inequality yields tnα−1 ≤ (C2 − C1 )nβ . So, t = O(n1+β−α ) and in obtaining H from G at most t · δnα−1 = O(nβ ) edges are removed. In fact, the above proof can be generalized a bit further as the following. Suppose F is a 13 family of bipartite graphs satisfying the property (P1) and the following property (P2). (P2). There exists some constants λ > 0 and 2 > α > β ≥ 1 such that z(n, F) = λnα + O(nβ ). Then Conjecture 1.1 holds for F in the following form: for any odd k ≥ k0 and sufficiently large n, any n-vertex F ∪ {Ck }-free extremal graph can be made bipartite by deleting a set of O(n1+β−α ) vertices, which together are incident to O(nβ ) edges. In particular, this also applies to F = {C4 , C6 , ..., C2ℓ } for ℓ ∈ {2, 3, 5}. 4. One could also prove our main theorems using the sparse regularity lemma ([22], [26]). However, the proofs would be more technical and would involve longer buildups. We chose to present a proof that avoids the use of sparse regularity. It seems, however, in order to make more progress on the original conjecture of Erdős and Simonovits (Conjecture 1.1), for instance to verify the conjecture for (α, β)-quasi-smooth families, sparse regularity lemma may still be an effective tool. This is because for (α, β)-quasi-smooth families F, like for (α, β)smooth families (see [1]), there is a transference of density from an F-free host graph to the corresponding cluster graph. References [1] P. Allen, P. Keevash, B. Sudakov and J. Verstraëte, Turán numbers of bipartite graphs plus an odd cycle, J. Combin. Theory Ser. B 106 (2014), 134-162. [2] N. Alon, L. Rónyai and T. Szabó, Norm-graphs: variations and applications, J. Combin. Theory Ser. B 76 (1999), 280-290. [3] N. Alon and J. H. Spencer, The Probabilistic Method, John Wiley & Sons, 3rd edition, 2016. [4] B. Andrásfai, P. Erdős and V.T. Sós, On the connection between chromatic number, maximal clique and minimal degree of a graph, Discrete Math. 8 (1974), 205-218. [5] J. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combin. Theory Ser. B 16 (1974), 97-105. [6] W. G. Brown, On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966), 281-285. [7] B. Bukh, Extremal graphs without exponentially-small bicliques, arXiv:2107.04167 [8] B. Bukh and M. Tait, Turán numbers of theta graphs, Combin. Probab. Comput. 29 (2020), 495-507. [9] D. Conlon, Graphs with few paths of prescribed length between any two vertices, Bull. Lond. Math. Soc. 51 (2019), 1015-1021. [10] P. Erdős and M. Simonovits, A limit theorem in graph theory, Studia Sci. Math. Hungar. 1 (1966), 51–57. [11] P. Erdős and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982), 275-288. [12] P. Erdős and H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1087–1091. [13] R. Faudree and M. Simonovits, On a class of degenerate extremal graph problems, Combinatorica, 3(1983), 83–93. 14 [14] Z. Füredi, An upper bound on Zarankiewicz’ problem, Combin. Probab. Comput. 1 (1996), 29-33. [15] Z. Füredi, New asymptotics for bipartite Turán numbers, J. Combin. Theory Ser. A, 75 (1996), 141-144. [16] Z. Füredi and M. Simonovits, The history of the degenerate (bipartite) extremal graph problems, Erdős centennial, Bolyai Soc. Math. Stud. 25, 169-264, János Bolyai Math. Soc., Budapest, 2013. See also arXiv:1306.5167. [17] S. Janson, T. Luczak and A. Rucinski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000. [18] T. Jiang and J. Ma, Cycles of given lengths in hypergraphs, J. Combin. Theory Ser. B 133 (2018), 54-77. [19] T. Jiang, J. Ma and L. Yepremyan, On Turán exponents of bipartite graphs, Combin. Probab. Comput., published online, https://doi.org/10.1017/S0963548321000341. See also arXiv:1806.02838. [20] P. Keevash, B. Sudakov and J. Verstraëte, On a conjecture of Erdős and Simonovits: Even cycles, Combinatorica 33 (2013), 699-732. [21] J. Kollár, L. Rónyai and T. Szabó, Norm-graphs and bipartite Turán numbers, Combinatorica 16 (1996), 399-406. [22] Y. Kohayakawa, Szemerédi’s regularity lemma for sparse graphs, in: F. Cucker, M. Shub (eds.), Foundations of Computational Mathematics, Springer-Verlag, Berlin, 1997, pp. 295-352. [23] T. Kövári, V.T. Sós and P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50-57. [24] S. Letzter, Hypergraphs with no tight cycles, arXiv:2106:12082 (2021). [25] A. Naor and J. Verstraëte, A note on bipartite graphs without 2k-cycles, Combin. Probab. Comput. 14 (2005), 845-849. [26] A. Scott, Szemerédi’s Regularity Lemma for matrices and sparse graphs, Combin. Probab. Comput. 20 (2011), 455-466. [27] J. Verstraëte, On arithmetic progressions of cycle lengths in graphs, Combin. Probab. Comput. 9 (2000), 369-373. 15