Infinite Ramsey-minimal graphs for star forests

TL;DR

This paper investigates the existence of Ramsey-minimal graphs for pairs of star forests, proving non-existence in cases involving infinite stars and constructing examples for finite cases, extending classical finite graph results to infinite graphs.

Contribution

It characterizes when Ramsey-minimal graphs exist for pairs of star forests, especially involving infinite stars, and constructs new examples for finite star forests, advancing the understanding of infinite Ramsey theory.

Findings

No Ramsey-minimal graphs exist for pairs with at least one infinite-star component.

Constructed a Ramsey-minimal graph for a finite star forest versus a subdivision graph.

Extended finite graph Ramsey results to the infinite case.

Abstract

For graphs $F$, $G$, and $H$, we write $F \to (G,H)$ if every red-blue coloring of the edges of $F$ produces a red copy of $G$ or a blue copy of $H$. The graph $F$ is said to be $(G,H)$-minimal if it is subgraph-minimal with respect to this property. The characterization problem for Ramsey-minimal graphs is classically done for finite graphs. In 2021, Barrett and the second author generalized this problem to infinite graphs. They asked which pairs $(G,H)$ admit a Ramsey-minimal graph and which ones do not. We show that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph. Also, we construct a Ramsey-minimal graph for a finite star forest versus a subdivision graph. This paper builds upon the results of Burr et al. in 1981 on Ramsey-minimal graphs for finite star forests.