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 0.82019, improving previous bounds.

Contribution

It establishes a new lower bound on the density of monochromatic infinite paths in 2-colored infinite complete graphs, surpassing earlier results.

Findings

Existence of a monochromatic infinite path with density ≥ 0.82019

Improved lower bound over previous results by Erdős-Galvin and DeBiasio-McKenney

Advances understanding of structure in infinite graph colorings

Abstract

For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. 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 Erd\H{o}s and Galvin, and of DeBiasio and McKenney.