## 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

#### Abstract

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.