On Ramsey-minimal infinite graphs
Jordan Mitchell Barrett, Valentino Vito

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 , , a common problem in Ramsey theory is to study graphs such that , i.e. every red-blue coloring of the edges of produces either a red or a blue . We generalize this study to infinite graphs , ; in particular, we want to determine if there is a minimal such . 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 to have a Ramsey-minimal graph. We use these to prove, for example, that if is an infinite star and , is a matching, then the pair admits no Ramsey-minimal graphs.
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