On Ramsey-minimal infinite graphs

TL;DR

This paper extends Ramsey theory to infinite graphs, exploring minimal graphs that guarantee monochromatic subgraphs under any edge coloring, and establishes conditions and examples where such minimal graphs do or do not exist.

Contribution

It generalizes finite Ramsey problems to infinite graphs, providing new conditions and results on the existence of Ramsey-minimal graphs for infinite structures.

Findings

No Ramsey-minimal graphs exist for the pair (infinite star, matching)

Compactness results relate infinite and finite Ramsey problems

Conditions identified for the existence of Ramsey-minimal graphs in infinite cases

Abstract

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \ge 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.