Ramsey upper density of infinite graph factors

TL;DR

This paper investigates the maximum upper density of monochromatic infinite subgraphs (F-factors) in 2-edge-colored complete graphs on natural numbers, providing new bounds that are sharp for certain graphs like cliques and cycles.

Contribution

It establishes a new lower bound for the upper density of monochromatic F-factors, matching known upper bounds for some graphs and improving bounds specifically for triangles.

Findings

New lower bounds for upper density of F-factors.

Bounds are sharp for cliques and odd cycles.

Improved lower bound for triangles to approximately 0.622.

Abstract

The study of upper density problems on Ramsey theory was initiated by Erd\H{o}s and Galvin in 1993. In this paper we are concerned with the following problem: given a fixed finite graph $F$, what is the largest value of $\lambda$ such that every 2-edge-coloring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite $F$-factor whose vertex set has upper density at least $\lambda$? Here we prove a new lower bound for this problem. For some choices of $F$, including cliques and odd cycles, this new bound is sharp, as it matches an older upper bound. For the particular case where $F$ is a triangle, we also give an explicit lower bound of $1-\frac{1}{\sqrt{7}}=0.62203\dots$, improving the previous best bound of 3/5.