Borel fractional colorings of Schreier graphs

TL;DR

This paper investigates the Borel fractional chromatic number of Schreier graphs associated with free parts of Bernoulli shifts of countable groups, establishing a key equality and asymptotic behavior for free groups.

Contribution

It proves that the Borel fractional chromatic number equals the reciprocal of the measurable independence number for these graphs, and determines its asymptotic value for free groups.

Findings

Borel fractional chromatic number equals 1 over the measurable independence number.

Asymptotic determination of the Borel fractional chromatic number for free groups.

Answers a question posed by Meehan regarding these graph invariants.

Abstract

Let $\Gamma$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $\Gamma$ (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.