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In this paper we consider coloring problems on graphs and other combinatorialstructures on standard Borel spaces. Our goal is to obtain sufficientconditions under which such colorings can be made well-behaved in the sense oftopology or measure. To this end, we show that such well-behaved colorings canbe produced using certain powerful techniques from finite combinatorics andcomputer science. First, we prove that efficient distributed coloringalgorithms (on finite graphs) yield well-behaved colorings of Borel graphs ofbounded degree; roughly speaking, deterministic algorithms produce Borelcolorings, while randomized algorithms give measurable and Baire-measurablecolorings. Second, we establish measurable and Baire-measurable versions of theSymmetric Lov\'{a}sz Local Lemma (under the assumption$\mathsf{p}(\mathsf{d}+1)^8 \leq 2^{-15}$, which is stronger than the standardLLL assumption $\mathsf{p}(\mathsf{d} + 1) \leq e^{-1}$ but still sufficientfor many applications). From these general results, we derive a number ofconsequences in descriptive combinatorics and ergodic theory.

Distributed Algorithms, the Lov\'{a}sz Local Lemma, and Descriptive Combinatorics