Ramsey-finiteness for graph pairs (G,H) holds precisely when the pair is a matching case, an odd-star-with-matchings case, or belongs to Faudree's star-forest family.
Faudree,Ramsey minimal graphs for forests, Ars Combin.31(1991), 117–124
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Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erd\H{o}s-Faudree-Schelp conjectures
Ramsey-finiteness for graph pairs (G,H) holds precisely when the pair is a matching case, an odd-star-with-matchings case, or belongs to Faudree's star-forest family.