Borel fractional colorings of Schreier graphs

Abstract

. Let Γ be a countable group and let G be the Schreier graph of the free part of the Bernoulli shift Γ ý 2 Γ (with respect to some finite subset F Ď Γ). 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 Γ is the free group, answering a question of Meehan.