arXiv subjects

Anton Bernshteyn

Publications and source records attributed to Anton Bernshteyn.

At least 37 records · Page 2Linked to original sources

arXiv2023

Counting colorings of triangle-free graphs

By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $Δ$ has chromatic number at most $(C+o(1))Δ/\log Δ$ for some universal constant $C > 0$. Using the entropy compression method, Molloy proved that one can in fact take $C = 1$. Here we show that for every $q \geq (1 + o(1))Δ/\log Δ$, the number $c(G,q)$ of proper $q$-colorings of $G$ satisfies $c(G, q) \,\geq\, \left(1 - \frac{1}{q}\right)^m ((1-o(1))q)^n$, where $n = |V(G)|$ and $m = |E(G)|$. Except for the $o(1)$ term, this lower bound is best possible as witnessed by random $Δ$-regular graphs. When $q = (1 + o(1)) Δ/\log Δ$, our result yields the inequality $c(G,q) \,\geq\, \exp\left((1 - o(1)) \frac{\log Δ}{2} n\right)$, which improves an earlier bound of Iliopoulos and yields the optimal value for the constant factor in the exponent. Furthermore, this result implies the optimal lower bound on the number of independent sets in $G$ due to Davies, Jenssen, Perkins, and Roberts. An important ingredient in our proof is the counting method that was recently developed by Rosenfeld. As a byproduct, we obtain an alternative proof of Molloy's bound $χ(G) \leq (1 + o(1))Δ/\log Δ$ using Rosenfeld's method in place of entropy compression (other proofs of Molloy's theorem using Rosenfeld's technique were given independently by Hurley and Pirot and Martinsson).

math.CO↗

arXiv2022

Weak degeneracy of graphs

Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every $d$-degenerate graph is also weakly $d$-degenerate. On the other hand, if $G$ is weakly $d$-degenerate, then $χ(G) \leq d + 1$ (and, moreover, the same bound holds for the list-chromatic and even the DP-chromatic number of $G$). It turns out that several upper bounds in graph coloring theory can be phrased in terms of weak degeneracy. For example, we show that planar graphs are weakly $4$-degenerate, which implies Thomassen's famous theorem that planar graphs are $5$-list-colorable. We also prove a version of Brooks's theorem for weak degeneracy: a connected graph $G$ of maximum degree $d \geq 3$ is weakly $(d-1)$-degenerate unless $G \cong K_{d + 1}$. (By contrast, all $d$-regular graphs have degeneracy $d$.) We actually prove an even stronger result, namely that for every $d \geq 3$, there is $ε> 0$ such that if $G$ is a graph of weak degeneracy at least $d$, then either $G$ contains a $(d+1)$-clique or the maximum average degree of $G$ is at least $d + ε$. Finally, we show that graphs of maximum degree $d$ and either of girth at least $5$ or of bounded chromatic number are weakly $(d - Ω(\sqrt{d}))$-degenerate, which is best possible up to the value of the implied constant.

math.CO↗

arXiv2022

Equivariant maps to subshifts whose points have small stabilizers

Let $Γ$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^Γ)$ to denote the free part of the Bernoulli shift action of $Γ$ on $k^Γ$. Seward and Tucker-Drob showed that there exists a free subshift $\mathcal{S} \subseteq \mathrm{Free}(2^Γ)$ such that every free Borel action of $Γ$ on a Polish space admits a Borel $Γ$-equivariant map to $\mathcal{S}$. Here we generalize this result as follows. Let $\mathcal{S}$ be a subshift of finite type (for example, $\mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $Γ$ with some finite number of colors). Suppose that $π\colon \mathrm{Free}(k^Γ) \to \mathcal{S}$ is a continuous $Γ$-equivariant map and let $\mathrm{Stab}(π)$ be the set of all group elements that fix every point in the image of $π$. Unless $π$ is constant, $\mathrm{Stab}(π)$ is a finite normal subgroup of $Γ$. We prove that there exists a subshift $\mathcal{S}' \subseteq \mathcal{S}$ such that the stabilizer of every point in $\mathcal{S}'$ is $\mathrm{Stab}(π)$ and every free Borel action of $Γ$ on a Polish space admits a Borel $Γ$-equivariant map to $\mathcal{S}'$. In particular, if the shift action of $Γ$ on the image of $π$ is faithful (i.e., if $\mathrm{Stab}(π)$ is trivial), then the subshift $\mathcal{S}'$ is free. As an application of this general result, we deduce that if $F$ is a finite symmetric subset of $Γ\setminus \{\mathbf{1}\}$ of size $|F| = d \geq 1$ and $\mathrm{Col}(F, d + 1) \subseteq (d+1)^Γ$ is the set of all proper $(d+1)$-colorings of the Cayley graph of $Γ$ corresponding to $F$, then there is a free subshift $\mathcal{S} \subseteq \mathrm{Col}(F, d+1)$ such that every free Borel action of $Γ$ on a Polish space admits a Borel $Γ$-equivariant map to $\mathcal{S}$.

math.DS↗

arXiv2022

Descriptive combinatorics and distributed algorithms

This is a draft of an article to appear in the October 2022 issue of the Notices of the AMS. In this survey article we explore a fascinating area called descriptive combinatorics and its recently discovered connections to distributed algorithms -- a fundamental part of computer science that is becoming increasingly important in the modern era of decentralized computation. In the first part of the article we give a brief introduction to some of the central notions and problems of descriptive combinatorics. The second part is devoted to an overview of some of the results concerning the interactions between descriptive combinatorics and distributed algorithms, as well as a few open problems. The article should be accessible to readers with little to no background in either descriptive set theory or computer science.

math.HO↗

arXiv2022

Searching for an Intruder on Graphs and Their Subdivisions

In this paper we analyze a variant of the pursuit-evasion game on a graph $G$ where the intruder occupies a vertex, is allowed to move to adjacent vertices or remain in place, and is 'invisible' to the searcher, meaning that the searcher operates with no knowledge of the position of the intruder. On each stage, the searcher is allowed to inspect an arbitrary set of $k$ vertices. The minimum $k$ for which the searcher can guarantee the capture of the intruder is called the inspection number of $G$. We also introduce and study the topological inspection number, a quantity that captures the limiting behavior of the inspection number under subdivisions of $G$. Our central theorem provides a full classification of graphs with topological inspection number up to $3$.

math.CO↗

arXiv2022

Borel fractional colorings of Schreier graphs

Let $Γ$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $Γ$ (with respect to some finite subset $F \subseteq Γ$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $Γ$ is the free group, answering a question of Meehan.

math.CO↗

arXiv2022

Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms

The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the Lovász Local Lemma (the LLL for short), which was introduced by Erdős and Lovász in the mid-1970s. Here we develop a version of the LLL that can be used to prove the existence of continuous colorings. We then give several applications in Borel and topological dynamics. * Seward and Tucker-Drob showed that every free Borel action $Γ\curvearrowright X$ of a countable group $Γ$ admits an equivariant Borel map $π\colon X \to Y$ to a free subshift $Y \subset 2^Γ$. We give a new simple proof of this result. * We show that for a countable group $Γ$, $\mathrm{Free}(2^Γ)$ is weakly contained, in the sense of Elek, in every free continuous action of $Γ$ on a zero-dimensional Polish space. This fact is analogous to the theorem of Abért and Weiss for probability measure-preserving actions and has a number of consequences in continuous combinatorics. In particular, we deduce that a coloring problem admits a continuous solution on $\mathrm{Free}(2^Γ)$ if and only if it can be solved on finite subgraphs of the Cayley graph of $Γ$ by an efficient deterministic distributed algorithm (this fact was also proved independently and using different methods by Seward). This establishes a formal correspondence between questions that have been studied independently in continuous combinatorics and in distributed computing.

math.CO↗

arXiv2022

Coloring graphs with forbidden bipartite subgraphs

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $Δ$, then $χ(G) \leq c_F Δ/ \logΔ$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $χ(G) \leq (t + o(1)) Δ/ \logΔ$ as $Δ\to \infty$. We improve this bound to $(1+o(1)) Δ/\log Δ$, making the constant factor independent of $t$. We further extend our result to the DP-coloring setting (also known as correspondence coloring), introduced by Dvořák and Postle.

math.CO↗

arXiv2021

Equitable Colorings of Borel Graphs

Hajnal and Szemerédi proved that if $G$ is a finite graph with maximum degree $Δ$, then for every integer $k \geqslant Δ+1$, $G$ has a proper coloring with $k$ colors in which every two color classes differ in size at most by $1$; such colorings are called equitable. We obtain an analog of this result for infinite graphs in the Borel setting. Specifically, we show that if $G$ is an aperiodic Borel graph of finite maximum degree $Δ$, then for each $k \geqslant Δ+ 1$, $G$ has a Borel proper $k$-coloring in which every two color classes are related by an element of the Borel full semigroup of $G$. In particular, such colorings are equitable with respect to every $G$-invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $Δ$-colorings of graphs with small average degree. Namely, we prove that if $Δ\geqslant 3$, $G$ does not contain a clique on $Δ+ 1$ vertices, and $μ$ is an atomless $G$-invariant probability measure such that the average degree of $G$ with respect to $μ$ is at most $Δ/5$, then $G$ has a $μ$-equitable $Δ$-coloring. As steps towards the proof of this result, we establish measurable and list coloring extensions of a strengthening of Brooks's theorem due to Kostochka and Nakprasit.

math.CO↗

arXiv2021

Local Coloring Problems on Smooth Graphs

We construct a smooth locally finite Borel graph $G$ and a local coloring problem $Π$ such that $G$ has a coloring $V(G) \to \mathbb{N}$ that solves $Π$, but no such coloring can be Borel.

math.LO↗

arXiv2021

A Fast Distributed Algorithm for $(Δ+ 1)$-Edge-Coloring

We present a deterministic distributed algorithm in the LOCAL model that finds a proper $(Δ+ 1)$-edge-coloring of an $n$-vertex graph of maximum degree $Δ$ in $\mathrm{poly}(Δ, \log n)$ rounds. This is the first nontrivial distributed edge-coloring algorithm that uses only $Δ+1$ colors (matching the bound given by Vizing's theorem). Our approach is inspired by the recent proof of the measurable version of Vizing's theorem due to Grebík and Pikhurko.

math.CO↗

arXiv2020

DP-Colorings of Hypergraphs

Classical problems in hypergraph coloring theory are to estimate the minimum number of edges, $m_2(r)$ (respectively, $m^\ast_2(r)$), in a non-$2$-colorable $r$-uniform (respectively, $r$-uniform and simple) hypergraph. The best currently known bounds are \[c \cdot \sqrt{r/\log r} \cdot 2^r \,\leqslant\, m_2(r) \,\leqslant\, C \cdot r^2 \cdot 2^r \qquad \text{and} \qquad c' \cdot r^{-\varepsilon} \cdot 4^r \,\leqslant\, m_2^\ast(r) \,\leqslant\, C' \cdot r^4 \cdot 4^r,\] for any fixed $\varepsilon > 0$ and some $c$, $c'$, $C$, $C' > 0$ (where $c'$ may depend on $\varepsilon$). In this paper we consider the same problems in the context of DP-coloring (also known as correspondence coloring), which is a generalization of list coloring introduced by Dvořák and Postle and related to local conflict coloring studied independently by Fraigniaud, Heinrich, and Kosowski. Let $\tilde{m}_2(r)$ (respectively, $\tilde{m}^\ast_2(r)$) denote the minimum number of edges in a non-$2$-DP-colorable $r$-uniform (respectively, $r$-uniform and simple) hypergraph. By definition, $\tilde{m}_2(r) \leqslant m_2(r)$ and $\tilde{m}^\ast_2(r)\leqslant m^\ast_2(r)$. While the proof of the bound $m^\ast_2(r) = Ω( r^{-3} 4^r)$ due to Erdős and Lovász also works for $\tilde{m}^\ast_2(r)$, we show that the trivial lower bound $\tilde{m}_2(r) \geqslant 2^{r-1}$ is asymptotically tight, i.e., $\tilde{m}_2(r) \leqslant (1 + o(1))2^{r-1}$. On the other hand, when $r \geqslant 2$ is even, we prove that the lower bound $\tilde{m}_2(r) \geqslant 2^{r-1}$ is not sharp, i.e., $\tilde{m}_2(r) \geqslant 2^{r-1}+1$. Whether this result holds for any odd values of $r$ remains an open problem. Nevertheless, we conjecture that the difference $\tilde{m}_2(r) - 2^{r-1}$ can be arbitrarily large.

math.CO↗

arXiv2020

A Short Proof of Bernoulli Disjointness via the Local Lemma

Recently, Glasner, Tsankov, Weiss, and Zucker showed that if $Γ$ is an infinite discrete group, then every minimal $Γ$-flow is disjoint from the Bernoulli shift $2^Γ$. Their proof is somewhat involved; in particular, it invokes separate arguments for different classes of groups. In this note, we give a short and self-contained proof of their result using purely combinatorial methods applicable to all groups at once. Our proof relies on the Lovász Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems.

math.DS↗

arXiv2020

Independent Sets in Algebraic Hypergraphs

In this paper we study hypergraphs definable in an algebraically closed field. Our goal is to show, in the spirit of the so-called transference principles in extremal combinatorics, that if a given algebraic hypergraph is "dense" in a certain sense, then a generic low-dimensional subset of its vertices induces a subhypergraph that is also "dense." (For technical reasons, we only consider low-dimensional subsets that are parameterized by rational functions.) Our proof approach is inspired by the hypergraph containers method, developed by Balogh, Morris, and Samotij and independently by Saxton and Thomason (although adapting this method to the algebraic setting presents some unique challenges that do not occur when working with finite hypergraphs). Along the way, we establish a natural generalization of the classical dimension of fibers theorem in algebraic geometry, which is interesting in its own right.

math.CO↗

arXiv2020

On Baire Measurable Colorings of Group Actions

The field of descriptive combinatorics investigates the question, to what extent can classical combinatorial results and techniques be made topologically or measure-theoretically well-behaved? This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action $α$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $α$ is smooth on a comeager set); this result confirms the "hardness" of finding a topologically well-behaved coloring. When $α$ is the shift action, we characterize the class of problems for which $α$ has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.

math.LO↗

arXiv2019

The Local Cut Lemma

The Lovász Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm, Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appealing to the LLL. The aim of this paper is to provide a generalization of the LLL which implies these new combinatorial results. This generalization, which we call the Local Cut Lemma, concerns a random cut in a directed graph with certain properties. Note that our result has a short probabilistic proof that does not use entropy compression. As a consequence, it not only shows that a certain probability is positive, but also gives an explicit lower bound for this probability. As an illustration, we present a new application (an improved lower bound on the number of edges in color-critical hypergraphs) as well as explain how to use the Local Cut Lemma to derive some of the results obtained previously using the entropy compression method.

math.CO↗

arXiv2019

Measurable versions of the Lovász Local Lemma and measurable graph colorings

In this paper we investigate the extent to which the Lovász Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lovász Local Lemma is used to produce a function $f \colon X \to Y$ with certain properties, where $X$ is some underlying combinatorial structure and $Y$ is a (typically finite) set. Can this function $f$ be chosen to be Borel or $μ$-measurable for some probability Borel measure $μ$ on $X$ (assuming that $X$ is a standard Borel space)? In the positive direction, we prove that if the set of constraints put on $f$ is, in a certain sense, "locally finite," then there is always a Borel choice for $f$ that is "$\varepsilon$-close" to satisfying these constraints, for any $\varepsilon > 0$. Moreover, if the combinatorial structure on $X$ is "induced" by the $[0;1]$-shift action of a countable group $Γ$, then, even without any local finiteness assumptions, there is a Borel choice for $f$ which satisfies the constraints on an invariant conull set (i.e., with $\varepsilon = 0$). A direct corollary of our results is an upper bound on the measurable chromatic number of the graph $G_n$ generated by the shift action of the free group $\mathbb{F}_n$ that is asymptotically tight up to a factor of at most $2$ (which answers a question of Lyons and Nazarov). On the other hand, our result for structures induced by measure-preserving group actions is, at least for amenable groups, sharp in the following sense: a probability measure-preserving action of a countably infinite amenable group satisfies the measurable version of the Lovász Local Lemma if and only if it admits a factor map to the $[0;1]$-shift action. To prove this, we combine the tools of the Ornstein--Weiss theory of entropy for actions of amenable groups with concepts from computability theory, specifically, Kolmogorov complexity.

math.CO↗

arXiv2019

Fractional DP-Colorings of Sparse Graphs

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. In this paper we introduce and study the fractional DP-chromatic number $χ_{DP}^\ast(G)$. We characterize all connected graphs $G$ such that $χ_{DP}^\ast(G) \leqslant 2$: they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt, the fractional list-chromatic number $χ_\ell^\ast(G)$ of any graph $G$ equals its fractional chromatic number $χ^\ast(G)$. This equality does not extend to fractional DP-colorings. Moreover, we show that the difference $χ^\ast_{DP}(G) - χ^\ast(G)$ can be arbitrarily large, and, furthermore, $χ^\ast_{DP}(G) \geq d/(2 \ln d)$ for every graph $G$ of maximum average degree $d \geq 4$. On the other hand, we show that this asymptotic lower bound is tight for a large class of graphs that includes all bipartite graphs as well as many graphs of high girth and high chromatic number.

math.CO↗