Density of monochromatic infinite paths

Abstract

For any subset $A\subseteq \mathbb{N}$, we define its upper density to be ${\mathrm{limsup}}_{n\to \infty }|A\cap \{1,\dots ,n\}|/n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9+\sqrt{17})/16\approx 0.82019$. This improves on results of Erdős and Galvin, and of DeBiasio and McKenney.

Accepted for publication in The Electronic Journal of Combinatorics