The Local Action Lemma

The Lov\'{a}sz 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.Comment: 18 pages; corrected typos, added references; added new Section 4.

Measurable versions of the Lov\'{a}sz Local Lemma and measurable graph colorings

In this paper we investigate the extent to which the Lov\'asz Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lov\'asz 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 $\mu$-measurable for some probability Borel measure $\mu$ 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 $\Gamma$, 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 Lov\'asz 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.Comment: 48 pages, 3 figure

The Local Cut Lemma

The Lov\'{a}sz 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.Comment: 19 pages, 2 figures. This updated version includes a simplified special case of the LCL (Theorem 3.1) which is sufficient for many applications and is, perhaps, somewhat more intuitiv

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 $\chi(G) \leq (1+o(1))\Delta(G)/\ln \Delta(G)$ for triangle-free graphs $G$, avoiding the technicalities of the entropy compression method and only using the usual "lopsided" Lov\'asz 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 Dvo\v{r}\'ak and Postle.Comment: 10 pages, 1 figure; v5: Minor changes following referees' suggestion

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 $\alpha$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $\alpha$ is smooth on a comeager set); this result confirms the "hardness" of finding a topologically well-behaved coloring. When $\alpha$ is the shift action, we characterize the class of problems for which $\alpha$ 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.Comment: 22 page

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 $a'(G)$. Fiam\v{c}ik and independently Alon, Sudakov, and Zaks conjectured that $a'(G) \leq \Delta(G)+2$, where $\Delta(G)$ denotes the maximum degree of $G$. The best known general bound is $a'(G)\leq 4(\Delta(G)-1)$ due to Esperet and Parreau. We apply a generalization of the Lov\'{a}sz Local Lemma to show that if $G$ contains no copy of a given bipartite graph $H$, then $a'(G) \leq 3\Delta(G)+o(\Delta(G))$. Moreover, for every $\varepsilon>0$, there exists a constant $c$ such that if $g(G)\geq c$, then $a'(G)\leq(2+\varepsilon)\Delta(G)+o(\Delta(G))$, where $g(G)$ denotes the girth of $G$.Comment: 12 pages, 2 figures. This version uses the Local Cut Lemma instead of the Local Action Lemm

The asymptotic behavior of the correspondence chromatic number

Alon proved that for any graph $G$, $\chi_\ell(G) = \Omega(\ln d)$, where $\chi_\ell(G)$ is the list chromatic number of $G$ and $d$ is the average degree of $G$. Dvo\v{r}\'{a}k 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 $\chi_c(G) = \Omega(d/\ln d)$, where $\chi_c(G)$ denotes the correspondence chromatic number of $G$. We also prove that for triangle-free $G$, $\chi_c(G) = O(\Delta/\ln \Delta)$, where $\Delta$ 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.Comment: 15 pages; a few typos fixe

On the number of edges in a graph with no (k+1)(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 $n\geq\frac{5k}{2}$.Comment: 8 pages; a few typos have been fixe

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

We present a deterministic distributed algorithm in the LOCAL model that finds a proper $(\Delta + 1)$-edge-coloring of an $n$-vertex graph of maximum degree $\Delta$ in $\mathrm{poly}(\Delta, \log n)$ rounds. This is the first nontrivial distributed edge-coloring algorithm that uses only $\Delta+1$ 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 Greb\'ik and Pikhurko.Comment: 24 pages, 9 figure

Building Large Free Subshifts Using the Local Lemma

Gao, Jackson, and Seward proved that every countably infinite group $\Gamma$ admits a nonempty free subshift $X \subseteq 2^\Gamma$. 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 $X \subseteq k^\Gamma$ of Hausdorff dimension and, if $\Gamma$ 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 Lov\'{a}sz Local Lemma, an important tool in probabilistic combinatorics, whose relevance to the problem of finding free subshifts was first recognized by Aubrun, Barbieri, and Thomass\'{e}.Comment: 13 page