Burr–Erdős–Faudree–Rousseau–Schelp one-sided Ramsey-infiniteness conjecture

Let $G$ and $H$ be graphs. A pair $(G,H)$ is Ramsey-infinite if it has infinitely many Ramsey-minimal graphs. An odd star is a star with an odd number of leaves, and a $K_2$-component is a connected component isomorphic to $K_2$. Burr–Erdős–Faudree–Rousseau–Schelp conjecture. The pair $(G,H)$ is Ramsey-infinite unless both $G$ and $H$ are odd stars or at least one of $G$ and $H$ contains a $K_2$-component. This is deliberately a one-sided assertion: the exceptional conditions are not claimed to characterize all Ramsey-finite pairs, since pairs with a $K_2$-component can still be Ramsey-infinite. The paper states that this conjecture is true.