Borel fractional perfect matchings in quasi-transitive amenable graphs

Abstract

We show that if a locally finite Borel graph with quasitransitive amenable components admits a fractional perfect matching, it will admit a Borel fractional perfect matching. In particular, if a countable amenable quasitransitive graph admits a fractional perfect matching then its Bernoulli graph admits a Borel fractional perfect matching.

18 pages, 5 figures; corrected reference