On Ramsey-minimal infinite graphs

• Main Authors: Barrett, Jordan Mitchell, Vito, Valentino
• Format: Journal Article
• Language: English
• Published: 11-03-2021
• Subjects: Mathematics - Combinatorics; Mathematics - Logic
• DOI: 10.48550/arxiv.2011.14074

The Electronic Journal of Combinatorics 28(1) (2021), #P1.46 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.