Upper density of monochromatic infinite paths

TL;DR

This paper proves that in any 2-coloring of an infinite complete graph, there exists a monochromatic infinite path with an upper density of at least approximately 0.87226, establishing the optimal bound and resolving a longstanding problem.

Contribution

It establishes the exact maximum upper density of monochromatic infinite paths in 2-colored infinite complete graphs, solving a problem posed by Erdős and Galvin.

Findings

Existence of a monochromatic infinite path with upper density at least 0.87226.

This bound is proven to be optimal.

Resolves a longstanding open problem in graph theory.

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 Erd\H{o}s and Galvin.