Borel fractional perfect matchings in quasi-transitive amenable graphs

TL;DR

This paper proves that certain amenable, quasi-transitive graphs with fractional perfect matchings also admit Borel fractional perfect matchings, linking graph properties with measure-theoretic matchings.

Contribution

It establishes that fractional perfect matchings in amenable, quasi-transitive graphs imply the existence of Borel fractional perfect matchings, extending previous results to a broader class of graphs.

Findings

Fractional perfect matchings imply Borel fractional perfect matchings in amenable graphs.

Countable amenable quasi-transitive graphs with fractional perfect matchings have Borel fractional perfect matchings.

Results connect graph matchings with measure-theoretic properties in Borel 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.