Combinatorics, Probability & Computing, Feb 3, 2023
Given a family F of bipartite graphs, the Zarankiewicz number z(m, n, F) is the maximum number of... more 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) = ρmn α−1 + O(n β). 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 k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least ρ(2n 5 + o(n)) α−1 is bipartite. In this paper, we strengthen their result by showing that for every real δ > 0, there exists k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least δn α−1 is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families F consisting of the single graph K s,t when t ≫ s. We also prove an analogous result for C 2ℓ-free graphs for every ℓ ≥ 2, which complements a result of Keevash, Sudakov and Verstraëte in [20].
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (... more The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets S in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of Füredi [12] and of Alon, Krivelevich, and Sudakov [2] showing that the maximum number of edges in an n-vertex graph not containing a fixed bipartite graph with maximum degree at most r on one side is O(n 2−1/r). This was recently extended by Grzesik, Janzer and Nagy [14] to the family of so-called (r, t)-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov [5], confirming a special case of a conjecture of Erdős and Simonovits and of Sidorenko, showing that if H is a bipartite graph that contains a vertex complete to the other part and G is a graph then the probability that the uniform random mapping from V (H) to V (G) is a homomorphism
In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjec... more In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjecture of Erdős on the number of C 2ℓ-free graphs on n vertices and gave new bounds on the Turán number of C 2ℓ 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), C 2ℓ) would also follow.
Given a family F of bipartite graphs, the Zarankiewicz number z(m, n, F) is the maximum number of... more 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) = ρmn α−1 + O(n β). 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 k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least ρ(2n 5 + o(n)) α−1 is bipartite. In this paper, we strengthen their result by showing that for every real δ > 0, there exists k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least δn α−1 is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families F consisting of the single graph K s,t when t ≫ s. We also prove an analogous result for C 2ℓ-free graphs for every ℓ ≥ 2, which complements a result of Keevash, Sudakov and Verstraëte in [20].
SIAM Journal on Discrete Mathematics, Aug 11, 2023
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (... more The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets S in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of Füredi [12] and of Alon, Krivelevich, and Sudakov [2] showing that the maximum number of edges in an n-vertex graph not containing a fixed bipartite graph with maximum degree at most r on one side is O(n 2−1/r). This was recently extended by Grzesik, Janzer and Nagy [14] to the family of so-called (r, t)-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov [5], confirming a special case of a conjecture of Erdős and Simonovits and of Sidorenko, showing that if H is a bipartite graph that contains a vertex complete to the other part and G is a graph then the probability that the uniform random mapping from V (H) to V (G) is a homomorphism
In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjec... more In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjecture of Erdős on the number of C 2ℓ-free graphs on n vertices and gave new bounds on the Turán number of C 2ℓ 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), C 2ℓ) would also follow.
Given a family ℱ of bipartite graphs, the Zarankiewicz number z(m,n,ℱ) is the maximum number of e... more 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)=ρ m n^α-1+O(n^β). Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, in <cit.> Allen, Keevash, Sudakov and Verstraëte proved that for any (α,β)-smooth family ℱ, there exists k_0 such that for all odd k≥ k_0 and sufficiently large n, any n-vertex ℱ∪{C_k}-free graph with minimum degree at least ρ(2n/5+o(n))^α-1 is bipartite. In this paper, we strengthen their result by showing that for every real δ>0, there exists k_0 such that for all odd k≥ k_0 and sufficiently large n, any n-vertex ℱ∪{C_k}-free graph with minimum degree at least δ n^α-1 is bipartite. Furthermore, our result holds u...
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (... more The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets S in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of Füredi [12] and of Alon, Krivelevich, and Sudakov [2] showing that the maximum number of edges in an n-vertex graph not containing a fixed bipartite graph with maximum degree at most r on one side is O(n). This was recently extended by Grzesik, Janzer and Nagy [14] to the family of so-called (r, t)-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov [5], confirming a special case of a conjecture of Erdős and Simonovits and of Sidorenko, showing that if H is a bipartite graph that contains a vertex complete to the other part and G is a graph then the probability that the uniform random mapping from V...
Given a family F of bipartite graphs, the Zarankiewicz number z(m,n,F) is the maximum number of e... more 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) = ρmn + O(n). 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 n-vertex F ∪ {Ck}-free graph with minimum degree at least ρ( 2n 5 + o(n)) is bipartite. 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 δn is bipartite. Furthermore, our result holds under ...
Combinatorics, Probability & Computing, Feb 3, 2023
Given a family F of bipartite graphs, the Zarankiewicz number z(m, n, F) is the maximum number of... more 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) = ρmn α−1 + O(n β). 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 k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least ρ(2n 5 + o(n)) α−1 is bipartite. In this paper, we strengthen their result by showing that for every real δ > 0, there exists k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least δn α−1 is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families F consisting of the single graph K s,t when t ≫ s. We also prove an analogous result for C 2ℓ-free graphs for every ℓ ≥ 2, which complements a result of Keevash, Sudakov and Verstraëte in [20].
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (... more The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets S in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of Füredi [12] and of Alon, Krivelevich, and Sudakov [2] showing that the maximum number of edges in an n-vertex graph not containing a fixed bipartite graph with maximum degree at most r on one side is O(n 2−1/r). This was recently extended by Grzesik, Janzer and Nagy [14] to the family of so-called (r, t)-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov [5], confirming a special case of a conjecture of Erdős and Simonovits and of Sidorenko, showing that if H is a bipartite graph that contains a vertex complete to the other part and G is a graph then the probability that the uniform random mapping from V (H) to V (G) is a homomorphism
In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjec... more In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjecture of Erdős on the number of C 2ℓ-free graphs on n vertices and gave new bounds on the Turán number of C 2ℓ 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), C 2ℓ) would also follow.
Given a family F of bipartite graphs, the Zarankiewicz number z(m, n, F) is the maximum number of... more 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) = ρmn α−1 + O(n β). 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 k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least ρ(2n 5 + o(n)) α−1 is bipartite. In this paper, we strengthen their result by showing that for every real δ > 0, there exists k 0 such that for all odd k ≥ k 0 and sufficiently large n, any n-vertex F ∪ {C k }-free graph with minimum degree at least δn α−1 is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families F consisting of the single graph K s,t when t ≫ s. We also prove an analogous result for C 2ℓ-free graphs for every ℓ ≥ 2, which complements a result of Keevash, Sudakov and Verstraëte in [20].
SIAM Journal on Discrete Mathematics, Aug 11, 2023
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (... more The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets S in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of Füredi [12] and of Alon, Krivelevich, and Sudakov [2] showing that the maximum number of edges in an n-vertex graph not containing a fixed bipartite graph with maximum degree at most r on one side is O(n 2−1/r). This was recently extended by Grzesik, Janzer and Nagy [14] to the family of so-called (r, t)-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov [5], confirming a special case of a conjecture of Erdős and Simonovits and of Sidorenko, showing that if H is a bipartite graph that contains a vertex complete to the other part and G is a graph then the probability that the uniform random mapping from V (H) to V (G) is a homomorphism
In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjec... more In a groundbreaking work utilizing the container method, Morris and Saxton [44] resolved a conjecture of Erdős on the number of C 2ℓ-free graphs on n vertices and gave new bounds on the Turán number of C 2ℓ 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), C 2ℓ) would also follow.
Given a family ℱ of bipartite graphs, the Zarankiewicz number z(m,n,ℱ) is the maximum number of e... more 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)=ρ m n^α-1+O(n^β). Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, in <cit.> Allen, Keevash, Sudakov and Verstraëte proved that for any (α,β)-smooth family ℱ, there exists k_0 such that for all odd k≥ k_0 and sufficiently large n, any n-vertex ℱ∪{C_k}-free graph with minimum degree at least ρ(2n/5+o(n))^α-1 is bipartite. In this paper, we strengthen their result by showing that for every real δ>0, there exists k_0 such that for all odd k≥ k_0 and sufficiently large n, any n-vertex ℱ∪{C_k}-free graph with minimum degree at least δ n^α-1 is bipartite. Furthermore, our result holds u...
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (... more The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets S in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of Füredi [12] and of Alon, Krivelevich, and Sudakov [2] showing that the maximum number of edges in an n-vertex graph not containing a fixed bipartite graph with maximum degree at most r on one side is O(n). This was recently extended by Grzesik, Janzer and Nagy [14] to the family of so-called (r, t)-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov [5], confirming a special case of a conjecture of Erdős and Simonovits and of Sidorenko, showing that if H is a bipartite graph that contains a vertex complete to the other part and G is a graph then the probability that the uniform random mapping from V...
Given a family F of bipartite graphs, the Zarankiewicz number z(m,n,F) is the maximum number of e... more 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) = ρmn + O(n). 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 n-vertex F ∪ {Ck}-free graph with minimum degree at least ρ( 2n 5 + o(n)) is bipartite. 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 δn is bipartite. Furthermore, our result holds under ...
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Papers by Sean Longbrake