arXiv subjects

Benny Sudakov

Publications and source records attributed to Benny Sudakov.

At least 109 records · Page 6Linked to original sources

arXiv2021

Proof of a conjecture on induced subgraphs of Ramsey graphs

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erdős, Faudree and Sós conjectured that in any C-Ramsey graph there are $Ω\left(n^{5/2}\right)$ induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture.

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arXiv2021

Short proofs of rainbow matching results

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that 'every edge coloured graph of a certain type contains a rainbow matching using every colour'. In this paper we introduce a versatile 'sampling trick', which allows us to obtain short proofs of old results as well as to solve asymptotically some well known conjectures. - We give a simple proof of Pokrovskiy's asymptotic version of the Aharoni-Berger conjecture with greatly improved error term. - We give the first asymptotic proof of the 'non-bipartite' Aharoni-Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky and Zerbib. - We give a very short asymptotic proof of Grinblat's conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat's problem as a function of edge multiplicity of the corresponding multigraph. - We give the first asymptotic proof of a 30 year old conjecture of Alspach.

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arXiv2021

Isomorphic Bisections of Cubic Graphs

Graph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively. In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs. Using probabilistic methods together with delicate recolouring arguments, we prove Ando's conjecture for large connected graphs.

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arXiv2021

Counting $H$-free orientations of graphs

In 1974, Erdős posed the following problem. Given an oriented graph $H$, determine or estimate the maximum possible number of $H$-free orientations of an $n$-vertex graph. When $H$ is a tournament, the answer was determined precisely for sufficiently large $n$ by Alon and Yuster. In general, when the underlying undirected graph of $H$ contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of $F$-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős's question. Moreover, we determine the answer exactly when $H$ is an odd cycle and $n$ is sufficiently large, answering a question of Araújo, Botler and Mota.

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arXiv2021

Which graphs can be counted in $C_4$-free graphs?

For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$ edges, then the density of $F$ in $G$, after a suitable normalization, is approximately at least the density of $F$ in an $ε$-regular approximation of $G$. In recent work, motivated by applications in extremal and additive combinatorics, we showed that $C_5$ has this property. Here we construct a family of graphs with the property.

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arXiv2021

Cycles of many lengths in Hamiltonian graphs

In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstraëte, who asked whether the regularity can be replaced with the weaker condition that the minimum degree is at least $3$. Despite attention from various researchers, until now, the best partial result towards both of these conjectures was a $\sqrt{n}$ lower bound on the number of cycle lengths. We resolve these conjectures asymptotically, by showing that the number of cycle lengths is at least $n^{1-o(1)}$.

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arXiv2021

New results for MaxCut in $H$-free graphs

The MaxCut problem asks for the size ${\rm mc}(G)$ of a largest cut in a graph $G$. It is well known that ${\rm mc}(G)\ge m/2$ for any $m$-edge graph $G$, and the difference ${\rm mc}(G)-m/2$ is called the surplus of $G$. The study of the surplus of $H$-free graphs was initiated by Erdős and Lovász in the 70s, who in particular asked what happens for triangle-free graphs. This was famously resolved by Alon, who showed that in the triangle-free case the surplus is $Ω(m^{4/5})$, and found constructions matching this bound. We prove several new results in this area. Firstly, we show that for every fixed odd $r\ge 3$, any $C_r$-free graph with $m$ edges has surplus $Ω_r\big(m^{\frac{r+1}{r+2}}\big)$. This is tight, as is shown by a construction of pseudorandom $C_r$-free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when $r$ is even. Secondly, generalizing the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs our bounds on the surplus are sharp. Thirdly, we prove that an $n$-vertex graph with few copies of $K_r$ and average degree $d$ has surplus $Ω_r(d^{r-1}/n^{r-3})$, which is tight when $d$ is close to $n$ provided that a conjectured dense pseudorandom $K_r$-free graph exists. This result is used to improve the best known lower bound (as a function of $m$) on the surplus of $K_r$-free graphs. Our proofs combine techniques from semidefinite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.

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arXiv2021

Ramsey properties of algebraic graphs and hypergraphs

One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on $N$ vertices. By the celebrated result of Erdős from 1947, the random graph on $N$ vertices with edge probability $1/2$, contains no clique or independent set larger than $2\log_2 N$, with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools. Say that an $r$-uniform hypergraph $\mathcal{H}$ is \emph{algebraic of complexity $(n,d,m)$} if the vertices of $\mathcal{H}$ are elements of $\mathbb{F}^{n}$ for some field $\mathbb{F}$, and there exist $m$ polynomials $f_1,\dots,f_m:(\mathbb{F}^{n})^{r}\rightarrow \mathbb{F}$ of degree at most $d$ such that the edges of $\mathcal{H}$ are determined by the zero-patterns of $f_1,\dots,f_m$. The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity $(n,d,m)$ has good Ramsey properties, then at least one of the parameters $n,d,m$ must be large. In 2001, Rónyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if $G$ is an algebraic graph of complexity $(n,d,m)$ on $N$ vertices, then either $G$ or its complement contains a complete balanced bipartite graph of size $Ω_{n,d,m}(N^{1/(n+1)})$. We extend this result by showing that such $G$ contains either a clique or an independent set of size $N^{Ω(1/ndm)}$ and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for $r$-uniform algebraic hypergraphs that are defined by a single polynomial, that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.

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arXiv2021

Flattening rank and its combinatorial applications

Given a $d$-dimensional tensor $T:A_1\times\dots\times A_d\rightarrow \mathbb{F}$ (where $\mathbb{F}$ is a field), the $i$-flattening rank of $T$ is the rank of the matrix whose rows are indexed by $A_{i}$, columns are indexed by $B_{i}=A_1\times\dots\times A_{i-1}\times A_{i+1}\times\dots\times A_{d}$ and whose entries are given by the corresponding values of $T$. The max-flattening rank of $T$ is defined as $\text{mfrank}(T)=\max_{i\in [d]}\text{frank}_{i}(T)$. A tensor $T:A^{d}\rightarrow\mathbb{F}$ is called semi-diagonal, if $T(a,\dots,a)\neq 0$ for every $a\in A$, and $T(a_{1},\dots,a_{d})=0$ for every $a_{1},\dots,a_{d}\in A$ that are all distinct. In this paper we prove that if $T:A^{d}\rightarrow\mathbb{F}$ is semi-diagonal, then $\text{mfrank}(T)\geq \frac{|A|}{d-1}$, and this bound is the best possible. We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an $r$-uniform multi-hypergraph $\mathcal{H}$ are colored with $z$ colors such that each colorclass is a matching of size $t$, then $\mathcal{H}$ contains a rainbow matching of size $t$ provided $z>(t-1)\binom{rt}{r}$. This improves previous results of Alon and Glebov, Sudakov and Szabó.

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arXiv2021

Powers of paths in tournaments

In this short note we prove that every tournament contains the $k$-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when $k=2$, showing that there is always a square of a directed path of length $\lceil 2n/3 \rceil-1$, which is best possible.

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arXiv2021

Long directed paths in Eulerian digraphs

An old conjecture of Bollobás and Scott asserts that every Eulerian directed graph with average degree $d$ contains a directed cycle of length at least $Ω(d)$. The best known lower bound for this problem is $Ω(d^{1/2})$ by Huang, Ma, Shapira, Sudakov and Yuster. They asked whether this estimate can be improved at least for directed paths instead of cycles and whether one can find a long path starting from any vertex if the host digraph is connected. In this paper we break the $\sqrt{d}$ barrier, showing how to find a path of length $Ω(d^{1/2+1/40})$ from any vertex of a connected Eulerian digraph.

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arXiv2020

Universal and unavoidable graphs

The Turán number $\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erdős asked which graphs $H$ with $e$ edges minimize $\text{ex}(n,H)$. They resolved this question asymptotically for most of the range of $e$ and asked to complete the picture. In this paper we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

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arXiv2020

Large induced matchings in random graphs

Given a large graph $H$, does the binomial random graph $G(n,p)$ contain a copy of $H$ as an induced subgraph with high probability? This classical question has been studied extensively for various graphs $H$, going back to the study of the independence number of $G(n,p)$ by Erdős and Bollobás, and Matula in 1976. In this paper we prove an asymptotically best possible result for induced matchings by showing that if $C/n\le p \le 0.99$ for some large constant $C$, then $G(n,p)$ contains an induced matching of order approximately $2\log_q(np)$, where $q= \frac{1}{1-p}$.

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arXiv2020

The intersection spectrum of 3-chromatic intersecting hypergraphs

For a hypergraph $H$, define its intersection spectrum $I(H)$ as the set of all intersection sizes $|E\cap F|$ of distinct edges $E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a $k$-uniform $3$-chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with $k$. Despite the problem being reiterated several times over the years by Erdős and other researchers, the lower bound of three intersection sizes has remarkably withstood any improvement until now. In this paper, we prove the Erdős-Lovász conjecture in a strong form by showing that there are at least $k^{1/2-o(1)}$ intersection sizes. Our proof consists of a delicate interplay between Ramsey type arguments and a density increment approach.

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arXiv2020

A Note on Powers of Paths in Tournaments

In this note we show that every tournament on $n$ vertices contains the $k$-th power of a directed path of length $n/2^{6k+7}$, which improves upon the recent bound of Scott and Korándi of $n/2^{2^{3k}}$. By doing so, we get an inverse exponential dependence on $k$, which is best possible as Yuster recently showed an upper bound of $kn/{2^{k/2}}$.

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arXiv2020

The extremal number of tight cycles

A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$. An old problem of V. Sós, also posed independently by J. Verstraëte, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. Our proof is based on finding robust expanders in the line graph of $\mathcal{H}$ together with certain density increment type arguments.

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arXiv2020

Infinite Sperner's theorem

One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice $2^{[n]}$ has size $Θ\big(\frac{2^n}{\sqrt{n}}\big)$. Motivated by an old problem of Erdős on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains. Using the well known Kraft's inequality for prefix codes, it is not difficult to show that infinite antichains should be "thinner" than the corresponding finite ones. More precisely, if $\mathcal{F}\subset 2^{\mathbb{N}}$ is an antichain, then $$\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap 2^{[n]}\big|\left(\frac{2^n}{n\log n}\right)^{-1}=0.$$ Our main result shows that this bound is essentially tight, that is, we construct an antichain $\mathcal{F}$ such that $$\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap 2^{[n]}\big|\left(\frac{2^n}{n\log^{C} n}\right)^{-1}>0$$ holds for some absolute constant $C>0$.

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arXiv2020

Short proofs of some extremal results III

We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.

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