Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms

02/17/2021 ∙ by Anton Bernshteyn, et al. ∙ 0

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↷ X of a countable group Γ admits an equivariant Borel map π X → Y to a free subshift Y ⊂ 2^Γ. We give a new simple proof of this result. * We show that for a countable group Γ, 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 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 Grebík, Jackson, Rozhoň, Seward, and Vidnyánszky). This establishes a formal correspondence between questions that have been studied independently in continuous combinatorics and in distributed computing.

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