Upper density of monochromatic infinite paths

Abstract

We prove that in every $2$-colouring of the edges of $K_{\mathbb{N}}$ there exists a monochromatic infinite path $P$ such that $V(P)$ has upper density at least $(12+\sqrt{8})/17\approx 0.87226$ and further show that this is best possible. This settles a problem of Erdős and Galvin.

16 pages, 2 figures