  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 Lov\'{a}sz 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.
