Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms

Abstract

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 Lov\'{a}sz Local Lemma (the LLL for short), which was introduced by Erd\H{o}s and Lov\'{a}sz 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 $\Gamma \curvearrowright X$ of a countable group $\Gamma$ admits an equivariant Borel map $\pi \colon X \to Y$ to a free subshift $Y \subset 2^\Gamma$. We give a new simple proof of this result. * We show that for a countable group $\Gamma$, $\mathrm{Free}(2^\Gamma)$ is weakly contained, in the sense of Elek, in every free continuous action of $\Gamma$ on a zero-dimensional Polish space. This fact is analogous to the theorem of Ab\'{e}rt 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 $\mathrm{Free}(2^\Gamma)$ if and only if it can be solved on finite subgraphs of the Cayley graph of $\Gamma$ 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.