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Benny Sudakov

Publications and source records attributed to Benny Sudakov.

At least 55 records · Page 3Linked to original sources
arXiv2024

Canonical Ramsey numbers of sparse graphs

The canonical Ramsey theorem of Erdős and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy of $H$. The least such $N$ is called the Erdős-Rado number of $H$, denoted by $ER(H)$. Erdős-Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős-Rado numbers of sparse graphs. For example, we prove that if $H$ has bounded degree, then $ER(H)$ is polynomial in $|V(H)|$ if $H$ is bipartite, but exponential in general. We also study the closely-related problem of constrained Ramsey numbers. For a given tree $S$ and given path $P_t$, we study the minimum $N$ such that every edge-coloring of $K_N$ contains a monochromatic copy of $S$ or a rainbow copy of $P_t$. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse-Ackermann type.

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arXiv2024

Chromatic number and regular subgraphs

In 1992, Erdős and Hajnal posed the following natural problem: Does there exist, for every $r\in \mathbb{N}$, an integer $F(r)$ such that every graph with chromatic number at least $F(r)$ contains $r$ edge-disjoint cycles on the same vertex set? We solve this problem in a strong form, by showing that there exist $n$-vertex graphs with fractional chromatic number $Ω\left(\frac{\log \log n}{\log \log \log n}\right)$ that do not even contain a $4$-regular subgraph. This implies that no such number $F(r)$ exists for $r\ge 2$. We show that assuming a conjecture of Harris, the bound on the fractional chromatic number in our result cannot be improved.

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arXiv2024

Induced subgraphs of $K_r$-free graphs and the Erdős--Rogers problem

For two graphs $F,H$ and a positive integer $n$, the function $f_{F,H}(n)$ denotes the largest $m$ such that every $H$-free graph on $n$ vertices contains an $F$-free induced subgraph on $m$ vertices. This function has been extensively studied in the last 60 years when $F$ and $H$ are cliques and became known as the Erdős-Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic study of this function in the case where $F$ is a general graph. Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there exists some $\varepsilon_F>0$ such that $f_{F,K_r}(n)=O(n^{1/2-\varepsilon_F})$. This result is tight in two ways. Firstly, it is no longer true if $F$ contains $K_{r-1}$ as a subgraph. Secondly, we show that for all $r\geq 4$ and $\varepsilon>0$, there exists a $K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=Ω(n^{1/2-\varepsilon})$. Along the way of proving this, we show in particular that for every graph $F$ with minimum degree $t$, we have $f_{F,K_4}(n)=Ω(n^{1/2-6/\sqrt{t}})$. This answers (in a strong form) another question of Mubayi and Verstraëte. Finally, we prove that there exist absolute constants $0<c<C$ such that for each $r\geq 4$, if $F$ is a bipartite graph with sufficiently large minimum degree, then $Ω(n^{\frac{c}{\log r}})\leq f_{F,K_r}(n)\leq O(n^{\frac{C}{\log r}})$. This shows that for graphs $F$ with large minimum degree, the behaviour of $f_{F,K_r}(n)$ is drastically different from that of the corresponding off-diagonal Ramsey number $f_{K_2,K_r}(n)$.

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arXiv2024

Counting subgraphs in locally dense graphs

A graph $G$ is said to be $p$-locally dense if every induced subgraph of $G$ with linearly many vertices has edge density at least $p$. A famous conjecture of Kohayakawa, Nagle, Rödl, and Schacht predicts that locally dense graphs have, asymptotically, at least as many copies of any fixed graph $H$ as are found in a random graph of edge density $p$. In this paper, we prove several results around the KNRS conjecture. First, we prove that certain natural gluing operations on $H$ preserve this property, thus proving the conjecture for many graphs $H$ for which it was previously unknown. Secondly, we study a stability version of this conjecture, and prove that for many graphs $H$, approximate equality is attained in the KNRS conjecture if and only if the host graph $G$ is quasirandom. Finally, we introduce a weakening of the KNRS conjecture, which requires the host graph to be nearly degree-regular, and prove this conjecture for a larger family of graphs. Our techniques reveal a surprising connection between these questions, semidefinite optimization, and the study of copositive matrices.

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arXiv2024

Approximate path decompositions of regular graphs

We show that the edges of any $d$-regular graph can be almost decomposed into paths of length roughly $d$, giving an approximate solution to a problem of Kotzig from 1957. Along the way, we show that almost all of the vertices of a $d$-regular graph can be partitioned into $n/(d+1)$ paths, asymptotically confirming a conjecture of Magnant and Martin from 2009.

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arXiv2024

The Helly number of Hamming balls and related problems

We prove the following variant of Helly's classical theorem for Hamming balls with a bounded radius. For $n>t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X^n$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X|>1$ and all $n>t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory.

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arXiv2024

Induced Ramsey problems for trees and graphs with bounded treewidth

The induced $q$-color size-Ramsey number $\hat{r}_{\text{ind}}(H;q)$ of a graph $H$ is the minimal number of edges a host graph $G$ can have so that every $q$-edge-coloring of $G$ contains a monochromatic copy of $H$ which is an induced subgraph of $G$. A natural question, which in the non-induced case has a very long history, asks which families of graphs $H$ have induced Ramsey numbers that are linear in $|H|$. We prove that for every $k,w,q$, if $H$ is an $n$-vertex graph with maximum degree $k$ and treewidth at most $w$, then $\hat{r}_{\text{ind}}(H;q) = O_{k,w,q}(n)$. This extends several old and recent results in Ramsey theory. Our proof is quite simple and relies upon a novel reduction argument.

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arXiv2024

Acyclic subgraphs of tournaments with high chromatic number

We prove that every $n$-vertex tournament $G$ has an acyclic subgraph with chromatic number at least $n^{5/9-o(1)}$, while there exists an $n$-vertex tournament $G$ whose every acyclic subgraph has chromatic number at most $n^{3/4+o(1)}$. This establishes in a strong form a conjecture of Nassar and Yuster and improves on another result of theirs. Our proof combines probabilistic and spectral techniques together with some additional ideas. In particular, we prove a lemma showing that every tournament with many transitive subtournaments has a large subtournament that is almost transitive. This may be of independent interest.

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arXiv2024

The growth rate of multicolor Ramsey numbers of $3$-graphs

The $q$-color Ramsey number of a $k$-uniform hypergraph $G,$ denoted $r(G;q)$, is the minimum integer $N$ such that any coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $G$. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behaviour of $r(G;q)$ for fixed $G$ and $q$ tending to infinity. In this paper we study this problem for $3$-uniform hypergraphs and determine the tower height of $r(G;q)$ as a function of $q$. More precisely, given a hypergraph $G$, we determine when $r(G; q)$ behaves polynomially, exponentially or double-exponentially in $q$. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.

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arXiv2024

Hamiltonicity of expanders: optimal bounds and applications

An $n$-vertex graph $G$ is a $C$-expander if $|N(X)|\geq C|X|$ for every $X\subseteq V(G)$ with $|X|< n/2C$ and there is an edge between every two disjoint sets of at least $n/2C$ vertices. We show that there is some constant $C>0$ for which every $C$-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in $(n,d,λ)$-graphs. This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications, including to the Hamiltonicity of random Cayley graphs.

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arXiv2024

Incidence bounds via extremal graph theory

The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no $s$ points are contained in the intersection of $s$ hyperplanes, the currently best known upper bound on the number of incidences of $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ is $$O_{d, s}((mn)^{1-1/(d+1)}+m+n).$$ This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers. In this paper, we propose a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in $\mathbb{R}^d$ for many interesting values of $m, n, d$, e.g. when $m=n$ and $d$ is odd. Moreover, in finite fields our bounds are sharp as a function of $m$ and $n$ in every dimension. We also study the size of the largest complete bipartite graph in point-hyperplane incidence graphs with a given number of edges and obtain optimal bounds as well. Additionally, we study point-variety incidences and unit-distance problem in finite fields, and give tight bounds for both problems under a similar non-degeneracy assumption. We also resolve Zarankiewicz type problems for algebraic graphs. Our proofs use tools such as induced Turán problems, VC-dimension theory, evasive sets and Hilbert polynomials. Also, we extend the celebrated result of Rónyai, Babai and Ganapathy on the number of zero-patterns of polynomials to the context of varieties, which might be of independent interest.

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arXiv2024

Point-variety incidences, unit distances and Zarankiewicz's problem for algebraic graphs

In this paper we study the number of incidences between $m$ points and $n$ varieties in $\mathbb{F}^d$, where $\mathbb{F}$ is an arbitrary field, assuming the incidence graph contains no copy of $K_{s,s}$. We also consider the analogous problem for algebraically defined graphs and unit distance graphs. First, we prove that if $\mathcal{P}$ is a set of $m$ points and $\mathcal{V}$ is a set of $n$ varieties in $\mathbb{F}^{D}$, each of dimension $d$ and degree at most $Δ$, and in addition the incidence graph is $K_{s,s}$-free, then the number of incidences satisfies $I(\mathcal{P}, \mathcal{V})\leq O_{d,Δ, s}(m^{\frac{d}{d+1}} n+m)$. This bound is tight when $s,Δ$ are sufficiently large with respect to $d$, with an appropriate choice of $\mathbb{F}=\mathbb{F}(m,n)$. We give two proofs of this upper bound, one based on the framework of the induced Turán problems and the other based on VC-dimension theory. In the second proof, we extend the celebrated result of Rónyai, Babai and Ganapathy on the number of zero-patterns of polynomials to the context of varieties, which might be of independent interest. We also resolve the problem of finding the maximum number of unit distances which can be spanned by a set of $n$ points $\mathcal{P}$ in $\mathbb{F^d}$ whose unit-distance graph is $K_{s, s}$-free, showing that it is $Θ_{d,s}(n^{2-\frac{1}{\lceil d/2\rceil +1}})$. Finally, we obtain tight bounds on the maximum number of edges of a $K_{s, s}$-free algebraic graph defined over a finite field, thus resolving the Zarankiewicz problem for this class of graphs.

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arXiv2024

Improved bounds for the Erdős-Rogers $(s,s+2)$-problem

For $2\leq s<t$, the Erdős-Rogers function $f_{s,t}(n)$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. There has been an extensive amount of work towards estimating this function, but until very recently only the case $t=s+1$ was well understood. A recent breakthrough of Mattheus and Verstraëte on the Ramsey number $r(4,k)$ states that $f_{2,4}(n)\leq n^{1/3+o(1)}$, which matches the known lower bound up to the $o(1)$ term. In this paper we build on their approach and generalize this result by proving that $f_{s,s+2}(n)\leq n^{\frac{2s-3}{4s-5}+o(1)}$ holds for every $s\geq 2$. This comes close to the best known lower bound, improves a substantial body of work and is the best that any construction of similar kind can give.

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arXiv2024

On the Turán number of the hypercube

In 1964, Erdős proposed the problem of estimating the Turán number of the $d$-dimensional hypercube $Q_d$. Since $Q_d$ is a bipartite graph with maximum degree $d$, it follows from results of Füredi and Alon, Krivelevich, Sudakov that $\mathrm{ex}(n,Q_d)=O_d(n^{2-1/d})$. A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm{ex}(n,Q_d)=o(n^{2-1/d})$. We obtain the first power-improvement for this old problem by showing that $\mathrm{ex}(n,Q_d)=O_d(n^{2-\frac{1}{d-1}+\frac{1}{(d-1)2^{d-1}}})$. This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any $n$-vertex, properly edge-coloured graph without a rainbow cycle has at most $O(n(\log n)^2)$ edges, improving the previous best bound of $n(\log n)^{2+o(1)}$ by Tomon. Furthermore, we show that any properly edge-coloured $n$-vertex graph with $ω(n\log n)$ edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight.

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arXiv2023

Difference-Isomorphic Graph Families

Many well-studied problems in extremal combinatorics deal with the maximum possible size of a family of objects in which every pair of objects satisfies a given restriction. One problem of this type was recently raised by Alon, Gujgiczer, Körner, Milojević and Simonyi. They asked to determine the maximum size of a family $\mathcal{G}$ of graphs on $[n]$, such that for every two $G_1,G_2 \in \mathcal{G}$, the graphs $G_1 \setminus G_2$ and $G_2 \setminus G_1$ are isomorphic. We completely resolve this problem by showing that this maximum is exactly $2^{\frac{1}{2}\big(\binom{n}{2} - \lfloor \frac{n}{2}\rfloor\big)}$ and characterizing all the extremal constructions. We also prove an analogous result for $r$-uniform hypergraphs.

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arXiv2023

Matrix discrepancy and the log-rank conjecture

Given an $m\times n$ binary matrix $M$ with $|M|=p\cdot mn$ (where $|M|$ denotes the number of 1 entries), define the discrepancy of $M$ as $\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot |Y|\big|$. Using semidefinite programming and spectral techniques, we prove that if $\mbox{rank}(M)\leq r$ and $p\leq 1/2$, then $$\mbox{disc}(M)\geq Ω(mn)\cdot \min\left\{p,\frac{p^{1/2}}{\sqrt{r}}\right\}.$$ We use this result to obtain a modest improvement of Lovett's best known upper bound on the log-rank conjecture. We prove that any $m\times n$ binary matrix $M$ of rank at most $r$ contains an $(m\cdot 2^{-O(\sqrt{r})})\times (n\cdot 2^{-O(\sqrt{r})})$ sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank $r$ is at most $O(\sqrt{r})$.

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arXiv2023

Positive discrepancy, MaxCut, and eigenvalues of graphs

The positive discrepancy of a graph $G$ of edge density $p=e(G)/\binom{v(G)}{2}$ is defined as $$\mbox{disc}^{+}(G)=\max_{U\subset V(G)}e(G[U])-p\binom{|U|}{2}.$$ In 1993, Alon proved (using the equivalent terminology of minimum bisections) that if $G$ is $d$-regular on $n$ vertices, and $d=O(n^{1/9})$, then $\mbox{disc}^{+}(G)=Ω(d^{1/2}n)$. We greatly extend this by showing that if $G$ has average degree $d$, then $\mbox{disc}^{+}(G)=Ω(d^{\frac{1}{2}}n)$ if $d\in [0,n^{\frac{2}{3}}]$, $Ω(n^2/d)$ if $d\in [n^{\frac{2}{3}},n^{\frac{4}{5}}]$, and $Ω(d^{\frac{1}{4}}n/\log n)$ if $d\in \left[n^{\frac{4}{5}},(\frac{1}{2}-\varepsilon)n\right]$. These bounds are best possible if $d\ll n^{3/4}$, and the complete bipartite graph shows that $\mbox{disc}^{+}(G)=Ω(n)$ cannot be improved if $d\approx n/2$. Our proofs are based on semidefinite programming and linear algebraic techniques. An interesting corollary of our results is that every $d$-regular graph on $n$ vertices with ${\frac{1}{2}+\varepsilon\leq \frac{d}{n}\leq 1-\varepsilon}$ has a cut of size $\frac{nd}{4}+Ω(n^{5/4}/\log n)$. This is not necessarily true without the assumption of regularity, or the bounds on $d$. The positive discrepancy of regular graphs is controlled by the second eigenvalue $λ_2$, as $\mbox{disc}^{+}(G)\leq \frac{λ_2}{2} n+d$. As a byproduct of our arguments, we present lower bounds on $λ_2$ for regular graphs, extending the celebrated Alon-Boppana theorem in the dense regime.

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