Density of monochromatic infinite subgraphs

TL;DR

This paper investigates the density of monochromatic infinite subgraphs in edge-colored infinite complete graphs, improving known bounds for paths and extending results to directed paths, trees, and locally finite graphs.

Contribution

It improves the lower bound for the density of monochromatic paths from 2/3 to 3/4 and establishes a tight bound of 2/3, also extending results to other graph types.

Findings

Monochromatic path density improved to at least 3/4.

Established a tight bound of 2/3 for path density.

Extended results to directed paths, trees, and locally finite graphs.

Abstract

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to wonder how "large" of a monochromatic copy of $G$ we can find with respect to some measure -- for instance, the density (or upper density) of the vertex set of $G$ in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked. In one of the few results in the area, Erd\H{o}s and Galvin proved that in every 2-coloring of $K_\mathbb{N}$, there exists a monochromatic path whose vertex set has upper density at least $2/3$, but it is not possible to do better than $8/9$. They also showed that for some sequence $\epsilon_n\to 0$, there exists a monochromatic path $P$ such that for infinitely many $n$, the set $\{1,2,...,n\}$ contains the first $(\frac{1}{3+\sqrt{3}}-\epsilon_n)n$ vertices of $P$, but it is not possible to do better than $2n/3$. We improve both results, in the former case achieving an upper density at least $3/4$ and in the latter case obtaining a tight bound of $2/3$. We also consider related problems for directed paths, trees (connected subgraphs), and a more general result which includes locally finite graphs for instance.