Measure-theoretic unfriendly colorings
Volume 226 / 2014
Fundamenta Mathematicae 226 (2014), 237-244 MSC: Primary 03E15, 28A05; Secondary 05C15, 37A20. DOI: 10.4064/fm226-3-3
We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the existence of such a partition, we show how it settles the dynamical analog of the problem (up to weak equivalence) for graphs induced by free, measure-preserving actions of groups with designated finite generating set. As a corollary, we obtain the existence of translation-invariant random unfriendly colorings of Cayley graphs of finitely generated groups.