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

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.