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arXiv · 2010.13633

Ramsey upper density of infinite graph factors

József Balogh·Ander Lamaison
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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.

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József Balogh, Ander Lamaison. 2020-10-26. Ramsey upper density of infinite graph factors. https://arxiv.org/abs/2010.13633

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