Proof of a Conjecture of Galvin

TL;DR

This paper proves a conjecture by Galvin, showing that in any finite coloring of pairs of real numbers, there exists a set homeomorphic to the rationals with pairs using at most two colors, using large cardinal assumptions.

Contribution

It verifies Galvin's conjecture from the 1970s and extends the result to a broad class of topological spaces beyond the reals.

Findings

Existence of a rational-homeomorphic set with at most two colors in any finite coloring

Use of large cardinal axioms in the proof

Extension of the result to a wide class of topological spaces

Abstract

We prove that if the set of unordered pairs of real numbers is colored by finitely many colors, there is a set of reals homeomorphic to the rationals whose pairs have at most two colors. Our proof uses large cardinals and it verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.