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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-3,
author = {Clinton T. Conley},
title = {Measure-theoretic unfriendly colorings},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {237-244},
zbl = {06314010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-3}
}
Clinton T. Conley. Measure-theoretic unfriendly colorings. Fundamenta Mathematicae, Tome 227 (2014) pp. 237-244. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-3/