Borel fractional colorings of Schreier graphs
Annales Henri Lebesgue, Volume 5 (2022), pp. 1151-1160.

Keywordsfractional coloring, Borel sets, Borel combinatorics, Schreier graph, group action, symbolic dynamics

### Abstract

Let $\Gamma$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift $\Gamma ↷{2}^{\Gamma }$ (with respect to some finite subset $F\subseteq \Gamma$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $\Gamma$ is the free group, answering a question of Meehan.

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