Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erd\H{o}s-Faudree-Schelp conjectures

TL;DR

This paper completely solves two longstanding conjectures on Ramsey-finiteness for graph pairs, characterizing when pairs are finite or infinite in terms of graph structure, and confirms the only finite pairs are specific known families.

Contribution

It proves two 1981 conjectures on Ramsey-finiteness preservation and provides a complete classification of finite graph pairs, refining previous characterizations.

Findings

Ramsey-finiteness is preserved by adjoining disjoint matchings.

Pairs are Ramsey-infinite unless both graphs are odd stars or one has a $K_2$ component.

The only finite pairs outside known families are from Faudree's star-forest family.

Abstract

For finite graphs $G$ and $H$, let $\RR(G,H)$ denote the isomorphism classes of Ramsey-minimal graphs for $(G,H)$. We prove two 1981 conjectures of Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp: Ramsey-finiteness is preserved by adjoining disjoint matchings, and $(G,H)$ is Ramsey-infinite unless both graphs are odd stars or one graph has a $K_2$ component. We also replace Burr's stronger 1979 survey characterization by the correct necessary-and-sufficient form: apart from the matching case and the odd-star-with-matchings case, the only additional finite pairs are Faudree's star-forest family.