arXiv subjects

Anton Bernshteyn

Publications and source records attributed to Anton Bernshteyn.

At least 19 recordsLinked to original sources

arXiv2026

Randomized Borel $(2d+1)$-coloring of digraphs

Let $G$ be a Borel digraph with maximum out-degree $d \in \mathbb{N}$. We show that $G$ admits a random Borel $(2d+1)$-coloring for which every edge is almost surely not monochromatic. This gives a simpler proof of a recent result of Pelayo-Gómez: such a graph $G$ admits a measurable proper $(2d+1)$-coloring with respect to any Borel probability measure on $V(G)$. Our proof is an adaptation of Pelayo-Gómez's proof to the randomized Borel setting.

math.LO↗

arXiv2026

Measurable matchings in unbalanced graphs

Let $G$ be a locally finite multigraph that is bipartite and "unbalanced," meaning that it has a nontrivial bipartition $(A,B)$ with $\mathrm{deg}(x) > \mathrm{deg}(y)$ for all $x \in A$ and $y \in B$. We explore matchings in such graphs through the lens of descriptive set theory. In particular, we show that when $G$ is Borel and $μ$ is a Borel probability measure on its vertex set, there is a Borel matching in $G$ that covers $μ$-almost every vertex in $A$. This was previously known only under the assumption that $μ$ is $G$-invariant, which we eliminate using a novel probabilistic approach. We also describe various extra conditions that imply the existence of a Borel matching covering every vertex in $A$. Along the way, we confirm a conjecture of the first and third named authors concerning the existence of Borel independent complete sections in Borel graphs of finite asymptotic separation index. In addition to their intrinsic interest, our results have applications to various other topics, such as edge-colorings, balanced orientations, and equidecomposition theory for group actions. For example, we show that the measurable edge-chromatic number of every Borel multigraph with finite maximum degree $Δ$ is at most $\lfloor\frac{3Δ}{2}\rfloor$, matching Shannon's optimal bound for finite multigraphs. Another example is that paradoxical Borel group actions with finite asymptotic separation index admit paradoxical decompositions with Borel pieces. This refines a result of Marks and Unger.

math.LO↗

arXiv2026

Borel Local Lemma: arbitrary random variables and limited exponential growth

The Lovász Local Lemma (the LLL for short) is a powerful tool in probabilistic combinatorics that is used to verify the existence of combinatorial objects with desirable properties. Recent years saw the development of various "constructive" versions of the LLL. A major success of this research direction is the Borel version of the LLL due to Csóka, Grabowski, Máthé, Pikhurko, and Tyros, which holds under a subexponential growth assumption. A drawback of their approach is that it only applies when the underlying random variables take values in a finite set. We present an alternative proof of a Borel version of the LLL that holds even if the underlying random variables are continuous and applies to dependency graphs of limited exponential growth.

math.CO↗

arXiv2026

On strongly and robustly critical graphs

In extremal combinatorics, it is common to focus on structures that are minimal with respect to a certain property. In particular, critical and list-critical graphs occupy a prominent place in graph coloring theory. Stiebitz, Tuza, and Voigt introduced strongly critical graphs, i.e., graphs that are $k$-critical yet $L$-colorable with respect to every non-constant assignment $L$ of lists of size $k-1$. Here we strengthen this notion and extend it to the framework of DP-coloring (or correspondence coloring) by defining robustly $k$-critical graphs as those that are not $(k-1)$-DP-colorable, but only due to the fact that $χ(G) = k$. We then seek general methods for constructing robustly critical graphs. Our main result is that if $G$ is a critical graph (with respect to ordinary coloring), then the join of $G$ with a sufficiently large clique is robustly critical; this is new even for strong criticality.

math.CO↗

arXiv2025

Embedding Borel graphs into grids of asymptotically optimal dimension

Let $G$ be a Borel graph all of whose finite subgraphs embed into the $d$-dimensional grid with diagonals. We show that then $G$ itself admits a Borel embedding into the Schreier graph of a free Borel action of $\mathbb Z^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$ is replaced by $O(ρ\log ρ)$, where $ρ$ is the polynomial growth rate of $G$.

math.CO↗

arXiv2025

Sunflowers in set systems with small VC-dimension

A family of $r$ distinct sets $\{A_1,\ldots, A_r\}$ is an $r$-sunflower if for all $1 \leqslant i < j \leqslant r$ and $1 \leqslant i' < j' \leqslant r$, we have $A_i \cap A_j = A_{i'} \cap A_{j'}$. Erdős and Rado conjectured in 1960 that every family $\mathcal{H}$ of $\ell$-element sets of size at least $K(r)^\ell$ contains an $r$-sunflower, where $K(r)$ is some function that depends only on $r$. We prove that if $\mathcal{H}$ is a family of $\ell$-element sets of VC-dimension at most $d$ and $|\mathcal{H}| > (C r (\log d+\log^\ast \ell))^\ell$ for some absolute constant $C > 0$, then $\mathcal{H}$ contains an $r$-sunflower. This improves a recent result of Fox, Pach, and Suk. When $d=1$, we obtain a sharp bound, namely that $|\mathcal{H}| > (r-1)^\ell$ is sufficient. Along the way, we establish a strengthening of the Kahn-Kalai conjecture for set families of bounded VC-dimension, which is of independent interest.

math.CO↗

arXiv2025

Flows with minimal subdynamics

Let $Γ$ be a countably infinite discrete group. A $Γ$-flow $X$ (i.e., a nonempty compact Hausdorff space equipped with a continuous action of $Γ$) is called $S$-minimal for a subset $S \subseteq Γ$ if the partial orbit $S \cdot x$ is dense for every point $x \in X$. We show that for any countable family $(S_n)_{n \in \mathbb{N}}$ of infinite subsets of $Γ$, there exists a free $Γ$-flow $X$ that is $S_n$-minimal for all $n \in \mathbb{N}$; additionally, $X$ can be taken to be a subflow of $2^Γ$. This vastly generalizes a result of Frisch, Seward, and Zucker, in which each $S_n$ is required to be a normal subgroup of $Γ$. As a corollary, we show that for a given Polish $Γ$-flow $X$, there exists a free $Γ$-flow $Y$ disjoint from $X$ in the sense of Furstenberg if and only if $X$ has no wandering points. This completes a line of inquiry started by Glasner, Tsankov, Weiss, and Zucker. As another application, we strengthen some of the results of Gao, Jackson, Krohne, and Seward on the structure of Borel complete sections. For example, we show that if $B$ is a Borel complete section in the free part of $2^Γ$, then every union of sufficiently many shifts of $B$ contains an orbit (previously, this was only known for open sets $B$). Although our main results are purely dynamical, their proofs rely on recently developed machinery from descriptive set-theoretic combinatorics, namely the asymptotic separation index introduced by Conley, Jackson, Marks, Seward, and Tucker-Drob and its links to the Lovász Local Lemma.

math.DS↗

arXiv2025

Fast algorithms for Vizing's theorem on bounded degree graphs

Vizing's theorem states that every graph $G$ of maximum degree $Δ$ can be properly edge-colored using $Δ+ 1$ colors. The fastest currently known $(Δ+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(m\sqrt{n})$, where $n :=|V(G)|$ and $m :=|E(G)|$. We investigate the case when $Δ$ is constant, i.e., $Δ= O(1)$. In this regime, the runtime of Sinnamon's algorithm is $O(n^{3/2})$, which can be improved to $O(n \log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O(n)$, which is obviously best possible. Prior to this work, no linear-time $(Δ+1)$-edge-coloring algorithm was known for any $Δ\geq 4$. Using some of the same ideas, we also develop new algorithms for $(Δ+1)$-edge-coloring in the $\mathsf{LOCAL}$ model of distributed computation. Namely, when $Δ$ is constant, we design a deterministic $\mathsf{LOCAL}$ algorithm with running time $\tilde{O}(\log^5 n)$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $O(\log ^2 n)$. Although our focus is on the constant $Δ$ regime, our results remain interesting for $Δ$ up to $\log^{o(1)} n$, since the dependence of their running time on $Δ$ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method.

cs.DS↗

arXiv2025

Weak Degeneracy of Planar Graphs

The weak degeneracy of a graph $G$ is a numerical parameter that was recently introduced by the first two authors with the aim of understanding the power of greedy algorithms for graph coloring. Every $d$-degenerate graph is weakly $d$-degenerate, but the converse is not true in general (for example, all connected $d$-regular graphs except cycles and cliques are weakly $(d-1)$-degenerate). If $G$ is weakly $d$-degenerate, then the list-chromatic number of $G$ is at most $d+1$, and the same upper bound holds for various other parameters such as the DP-chromatic number and the paint number. Here we rectify a mistake in a paper of the first two authors and give a correct proof that planar graphs are weakly $4$-degenerate, strengthening the famous result of Thomassen that planar graphs are $5$-list-colorable.

math.CO↗

arXiv2025

Coloring graphs with forbidden almost bipartite subgraphs

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $χ(G) \leq (c(F) + o(1)) Δ/ \logΔ$ whenever $G$ is an $F$-free graph of maximum degree $Δ$. The largest class of connected graphs $F$ for which this conjecture has been verified so far, by Alon, Krivelevich, and Sudakov themselves, comprises the almost bipartite graphs (i.e., subgraphs of the complete tripartite graph $K_{1,t,t}$ for some $t \in \mathbb{N}$). However, the optimal value for $c(F)$ remains unknown even for such graphs. Bollobás showed, using random regular graphs, that $c(F) \geq 1/2$ when $F$ contains a cycle. On the other hand, Davies, Kang, Pirot, and Sereni recently established an upper bound of $c(K_{1,t,t}) \leq t$. We improve this to a uniform constant, showing $c(F) \leq 4$ for every almost bipartite graph $F$. This surprisingly makes the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of our bound in the setting of DP-coloring (also known as correspondence coloring) and consider some algorithmic consequences of our results.

math.CO↗

arXiv2025

Separating complexity classes of LCL problems on grids

We study the complexity of locally checkable labeling (LCL) problems on $\mathbb{Z}^n$ from the point of view of descriptive set theory, computability theory, and factors of i.i.d. Our results separate various complexity classes that were not previously known to be distinct and serve as counterexamples to a number of natural conjectures in the field.

math.LO↗

arXiv2025

Large-scale geometry of Borel graphs of polynomial growth

We study graphs of polynomial growth from the perspective of asymptotic geometry and descriptive set theory. The starting point of our investigation is a theorem of Krauthgamer and Lee who showed that every connected graph of polynomial growth admits an injective contraction mapping to $(\mathbb Z^n, \|\cdot\|_\infty)$ for some $n\in\mathbb N$. We strengthen and generalize this result in a number of ways. In particular, answering a question of Papasoglu, we construct coarse embeddings from graphs of polynomial growth to $\mathbb Z^n$. Moreover, we only require $n$ to be linear in the asymptotic polynomial growth rate of the graph; this confirms a conjecture of Levin and Linial, London, and Rabinovich "in the asymptotic sense." (The exact form of the conjecture was refuted by Krauthgamer and Lee.) All our results are proved for Borel graphs, which allows us to settle a number of problems in descriptive combinatorics. Roughly, we prove that graphs generated by free Borel actions of $\mathbb Z^n$ are universal for the class of Borel graphs of polynomial growth. This provides a general method for extending results about $\mathbb Z^n$-actions to all Borel graphs of polynomial growth. For example, an immediate consequence of our main result is that all Borel graphs of polynomial growth are hyperfinite, which answers a well-known question in the area. As another illustration, we show that Borel graphs of polynomial growth support a certain combinatorial structure called toast. An important technical tool in our arguments is the notion of padded decomposition from computer science, which is closely related to the concept of asymptotic dimension due to Gromov. Along the way we find an alternative, probabilistic proof of a theorem of Papasoglu that graphs of asymptotic polynomial growth rate $ρ<\infty$ have asymptotic dimension at most $ρ$ and establish the same bound in the Borel setting.

math.CO↗

arXiv2025

Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lovász Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index.

math.LO↗

arXiv2025

DP-Coloring of Graphs from Random Covers

DP-coloring (also called correspondence coloring) of graphs is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in $2015$. Intuitively, DP-coloring generalizes list coloring by allowing the colors that are identified as the same to vary from edge to edge. Formally, DP-coloring of a graph $G$ is equivalent to an independent transversal in an auxiliary structure called a DP-cover of $G$. In this paper, we introduce the notion of random DP-covers and study the behavior of DP-coloring from such random covers. We prove a series of results about the probability that a graph is or is not DP-colorable from a random cover. These results support the following threshold behavior on random $k$-fold DP-covers as $ρ\to\infty$ where $ρ$ is the maximum density of a graph: graphs are non-DP-colorable with high probability when $k$ is sufficiently smaller than $ρ/\lnρ$, and graphs are DP-colorable with high probability when $k$ is sufficiently larger than $ρ/\lnρ$. Our results depend on $ρ$ growing fast enough and imply a sharp threshold for dense enough graphs. For sparser graphs, we analyze DP-colorability in terms of degeneracy. We also prove fractional DP-coloring analogs to these results.

math.CO↗

arXiv2025

A linear-time algorithm for $(1+ε)Δ$-edge-coloring

We present a randomized algorithm that, given a constant $ε> 0$, outputs a proper $(1+ε)Δ$-edge-coloring of an $m$-edge simple graph $G$ of maximum degree $Δ\geq 1/ε$ in $O(m)$ time with high probability. This is the first linear-time algorithm for this problem covering the full range of possible values of $Δ$. Indeed, even for edge-coloring with $2Δ- 1$ colors (i.e., meeting the "greedy" bound), no such linear-time algorithm has been previously known.

cs.DS↗

arXiv2024

Borel line graphs

We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the 9 finite graphs from the classical result of Beineke together with a 10th infinite graph associated to the equivalence relation $\mathbb{E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman--Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

math.LO↗

arXiv2024

Borel Vizing's Theorem for Graphs of Subexponential Growth

We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $Δ(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $Δ(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$.

math.CO↗

arXiv2023

Distributed Algorithms, the Lovász Local Lemma, and Descriptive Combinatorics

In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or measure. To this end, we show that such well-behaved colorings can be produced using certain powerful techniques from finite combinatorics and computer science. First, we prove that efficient distributed coloring algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of bounded degree; roughly speaking, deterministic algorithms produce Borel colorings, while randomized algorithms give measurable and Baire-measurable colorings. Second, we establish measurable and Baire-measurable versions of the Symmetric Lovász Local Lemma (under the assumption $\mathsf{p}(\mathsf{d}+1)^8 \leq 2^{-15}$, which is stronger than the standard LLL assumption $\mathsf{p}(\mathsf{d} + 1) \leq e^{-1}$ but still sufficient for many applications). From these general results, we derive a number of consequences in descriptive combinatorics and ergodic theory.

math.CO↗