arXiv subjects

Anton Bernshteyn

Publications and source records attributed to Anton Bernshteyn.

52 records · Page 3Linked to original sources

arXiv2019

Improved Lower Bound for Difference Bases

A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A - A \supseteq \{1, \ldots, n\}$. Rédei and Rényi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$ for some positive constant $c$. The best previously known lower bound on $c$ is $c \geqslant 1.5602\ldots$, which was obtained by Leech using a version of an earlier argument due to Rédei and Rényi. In this note we use Fourier-analytic tools to show that the Leech--Rédei--Rényi lower bound is not sharp.

math.CO↗

arXiv2019

Multiplication of Weak Equivalence Classes May Be Discontinuous

For a countably infinite group $Γ$, let $\mathcal{W}_Γ$ denote the space of all weak equivalence classes of measure-preserving actions of $Γ$ on atomless standard probability spaces, equipped with the compact metrizable topology introduced by Abért and Elek. There is a natural multiplication operation on $\mathcal{W}_Γ$ (induced by taking products of actions) that makes $\mathcal{W}_Γ$ an Abelian semigroup. Burton, Kechris, and Tamuz showed that if $Γ$ is amenable, then $\mathcal{W}_Γ$ is a topological semigroup, i.e., the product map $\mathcal{W}_Γ\times \mathcal{W}_Γ\to \mathcal{W}_Γ\colon (\mathfrak{a}, \mathfrak{b}) \mapsto \mathfrak{a} \times \mathfrak{b}$ is continuous. In contrast to that, we prove that if $Γ$ is a Zariski dense subgroup of $\mathrm{SL}_d(\mathbb{Z})$ for some $d \geqslant 2$ (for instance, if $Γ$ is a non-Abelian free group), then multiplication on $\mathcal{W}_Γ$ is discontinuous, even when restricted to the subspace $\mathcal{FW}_Γ$ of all free weak equivalence classes.

math.DS↗

arXiv2019

Ergodic Theorems for the Shift Action and Pointwise Versions of The Abért--Weiss Theorem

Let $Γ$ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action $Γ\curvearrowright (X, μ)$ and a map $f \in L^1(X, μ)$, and to compare the global average $\int f \,\mathrm{d}μ$ of $f$ to the pointwise averages $|D|^{-1} \sum_{δ\in D} f(δ\cdot x)$, where $x \in X$ and $D$ is a nonempty finite subset of $Γ$. The basic hope is that, when $D$ runs over a suitably chosen infinite sequence, these pointwise averages should converge to the global value for $μ$-almost all $x$. In this paper we prove several results that refine the above basic paradigm by uniformly controlling the averages over specific sets $D$ rather than considering their limit as $|D| \to \infty$. Our results include ergodic theorems for the Bernoulli shift action $Γ\curvearrowright ([0;1]^Γ, λ^Γ)$ and strengthenings of the theorem of Abért and Weiss that the shift is weakly contained in every free p.m.p. action of $Γ$. In particular, we establish a purely Borel version of the Abért--Weiss theorem for finitely generated groups of subexponential growth. The central role in our arguments is played by the recently introduced measurable versions of the Lovász Local Lemma, due to the current author and to Csóka, Grabowski, Máthé, Pikhurko, and Tyros.

math.DS↗

arXiv2019

Building Large Free Subshifts Using the Local Lemma

Gao, Jackson, and Seward proved that every countably infinite group $Γ$ admits a nonempty free subshift $X \subseteq 2^Γ$. Here we strengthen this result by showing that free subshifts can be "large" in various senses. Specifically, we prove that for any $k \geqslant 2$ and $h < \log_2 k$, there exists a free subshift $X \subseteq k^Γ$ of Hausdorff dimension and, if $Γ$ is sofic, entropy at least $h$, answering two questions attributed by Gao, Jackson, and Seward to Juan Souto. Furthermore, we establish a general lower bound on the largest "size" of a free subshift $X'$ contained inside a given subshift $X$. A central role in our arguments is played by the Lovász Local Lemma, an important tool in probabilistic combinatorics, whose relevance to the problem of finding free subshifts was first recognized by Aubrun, Barbieri, and Thomassé.

math.DS↗

arXiv2018

A short nonalgorithmic proof of the containers theorem for hypergraphs

Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogs of a variety of classical problems from combinatorics and number theory. The previously known proofs of the containers theorem use the so-called scythe algorithm---an iterative procedure that runs through the vertices of the hypergraph. (Saxton and Thomason have also proposed an alternative, randomized construction in the case of simple hypergraphs.) Here we present the first known deterministic proof of the containers theorem that is not algorithmic, i.e., it does not involve an iterative process. Our proof is less than 4 pages long while being entirely self-contained and conceptually transparent. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. Before presenting the proof in full detail, we include a one-page informal outline that refers to this notion of dimension and summarizes the essence of the argument.

math.CO↗

arXiv2018

The Johansson--Molloy Theorem for DP-Coloring

The aim of this note is twofold. On the one hand, we present a streamlined version of Molloy's new proof of the bound $χ(G) \leq (1+o(1))Δ(G)/\ln Δ(G)$ for triangle-free graphs $G$, avoiding the technicalities of the entropy compression method and only using the usual "lopsided" Lovász Local Lemma (albeit in a somewhat unusual setting). On the other hand, we extend Molloy's result to DP-coloring (also known as correspondence coloring), a generalization of list coloring introduced recently by Dvořák and Postle.

math.CO↗

arXiv2018

New Bounds for the Acyclic Chromatic Index

An edge coloring of a graph $G$ is called an acyclic edge coloring if it is proper and every cycle in $G$ contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of $G$ is called the acyclic chromatic index of $G$ and is denoted by $a'(G)$. Fiamčik and independently Alon, Sudakov, and Zaks conjectured that $a'(G) \leq Δ(G)+2$, where $Δ(G)$ denotes the maximum degree of $G$. The best known general bound is $a'(G)\leq 4(Δ(G)-1)$ due to Esperet and Parreau. We apply a generalization of the Lovász Local Lemma to show that if $G$ contains no copy of a given bipartite graph $H$, then $a'(G) \leq 3Δ(G)+o(Δ(G))$. Moreover, for every $\varepsilon>0$, there exists a constant $c$ such that if $g(G)\geq c$, then $a'(G)\leq(2+\varepsilon)Δ(G)+o(Δ(G))$, where $g(G)$ denotes the girth of $G$.

math.CO↗

arXiv2017

Sharp Dirac's Theorem for DP-Critical Graphs

Correspondence coloring, or DP-coloring, is a generalization of list coloring introduced recently by Dvořák and Postle. In this paper we establish a version of Dirac's theorem on the minimum number of edges in critical graphs in the framework of DP-colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz on classifying list-critical graphs that satisfy Dirac's bound with equality.

math.CO↗

arXiv2017

On differences between DP-coloring and list coloring

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all of them do. In this note we describe some unusual properties of DP-coloring that set it aside from list coloring. In particular, we give an example of a planar bipartite graph with DP-chromatic number $4$ and prove that the edge-DP-chromatic number of a $d$-regular graph with $d\geq 2$ is always at least $d+1$.

math.CO↗

arXiv2017

DP-colorings of graphs with high chromatic number

DP-coloring is a generalization of list coloring introduced recently by Dvořák and Postle. We prove that for every $n$-vertex graph $G$ whose chromatic number $χ(G)$ is "close" to $n$, the DP-chromatic number of $G$ equals $χ(G)$. "Close" here means $χ(G)\geq n-O(\sqrt{n})$, and we also show that this lower bound is best possible (up to the constant factor in front of $\sqrt{n}$), in contrast to the case of list coloring.

math.CO↗

arXiv2017

Free Subshifts with Invariant Measures from the Lovász Local Lemma

Gao, Jackson, and Seward (see arXiv:1201.0513) proved that every countably infinite group $Γ$ admits a nonempty free subshift $X \subseteq \{0,1\}^Γ$. Furthermore, a theorem of Seward and Tucker-Drob (see arXiv:1402.4184) implies that every countably infinite group $Γ$ admits a free subshift $X \subseteq \{0,1\}^Γ$ that supports an invariant probability measure. Aubrun, Barbieri, and Thomassé (see arXiv:1507.03369) used the Lovász Local Lemma to give a short alternative proof of the Gao--Jackson--Seward theorem. Recently, Elek (see arXiv:1702.01631) followed another approach involving the Lovász Local Lemma to obtain a different proof of the existence of free subshifts with invariant probability measures for finitely generated sofic groups. Using the measurable version of the Lovász Local Lemma for shift actions established by the author (see arXiv:1604.07349), we give a short alternative proof of the existence of such subshifts for arbitrary groups. Moreover, we can find such subshifts in any nonempty invariant open set.

math.DS↗

arXiv2016

On DP-coloring of graphs and multigraphs

While solving a question on list coloring of planar graphs, Dvořák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph $G$ reduces the problem of finding a coloring of $G$ from a given list $L$ to the problem of finding a "large" independent set in an auxiliary graph $H(G,L)$ with vertex set $\{(v,c)\,: \, v\in V(G) \text{ and } {c\in L(v)} \}$. It is similar to the old reduction by Plesnevič and Vizing of the $k$-coloring problem to the problem of finding an independent set of size $|V(G)|$ in the Cartesian product $G\square K_k$. Some properties of the DP-chromatic number $χ_{DP}(G)$ resemble the properties of the list chromatic number $χ_{\ell}(G)$ but some differ quite a lot. It is always the case that $χ_{DP}(G)\geq χ_{\ell}(G)$. The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erdős, Rubin, and Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai's Theorem on the minimum number of edges in $n$-vertex graphs critical with respect to DP-coloring.

math.CO↗

arXiv2016

The asymptotic behavior of the correspondence chromatic number

Alon proved that for any graph $G$, $χ_\ell(G) = Ω(\ln d)$, where $χ_\ell(G)$ is the list chromatic number of $G$ and $d$ is the average degree of $G$. Dvořák and Postle recently introduced a generalization of list coloring, which they called correspondence coloring. We establish an analogue of Alon's result for correspondence coloring; namely, we show that $χ_c(G) = Ω(d/\ln d)$, where $χ_c(G)$ denotes the correspondence chromatic number of $G$. We also prove that for triangle-free $G$, $χ_c(G) = O(Δ/\ln Δ)$, where $Δ$ is the maximum degree of $G$ (this is a generalization of Johansson's result about list colorings). This implies that the correspondence chromatic number of a regular triangle-free graph is, up to a constant factor, determined by its degree.

math.CO↗

arXiv2016

Regular colorings and factors of regular graphs

An $(r-1,1)$-coloring of an $r$-regular graph $G$ is an edge coloring such that each vertex is incident to $r-1$ edges of one color and $1$ edge of a different color. In this paper, we completely characterize all $4$-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a $(3,1)$-coloring. An $\{r-1,1\}$-factor of an $r$-regular graph is a spanning subgraph in which each vertex has degree either $r-1$ or $1$. We prove various conditions that that must hold for any vertex-minimal $5$-regular pseudographs without $(4,1)$-colorings or without $\{4,1\}$-factors. Finally, for each $r\geq 6$ we construct graphs that are not $(r-1,1)$-colorable and, more generally, are not $(r-t,t)$-colorable for small $t$.

math.CO↗

arXiv2015

On the number of edges in a graph with no $(k+1)$-connected subgraphs

Mader proved that for $k\geq 2$ and $n\geq 2k$, every $n$-vertex graph with no $(k+1)$-connected subgraphs has at most $(1+\frac{1}{\sqrt{2}})k(n-k)$ edges. He also conjectured that for $n$ large with respect to $k$, every such graph has at most $\frac{3}{2}\left(k - \frac{1}{3}\right)(n-k)$ edges. Yuster improved Mader's upper bound to $\frac{193}{120}k(n-k)$ for $n\geq\frac{9k}{4}$. In this note, we make the next step towards Mader's Conjecture: we improve Yuster's bound to $\frac{19}{12}k(n-k)$ for $n\geq\frac{5k}{2}$.

math.CO↗

arXiv2015

The Local Action 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. We provide a general statement that implies both these new results and the LLL itself.

math.CO↗