Ramsey-minimal graphs for star-forests

It is shown that if G and H are star-forests with no single edge stars, then (G, H) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges . Further (5,,, U kS 1 , S, U 1S T ) is Ramsey-finite when m and n are odd, where S, denotes a star with i edges . In general, for G and H star-forests, (G U kS i , H U lS,) can be shown to be Ramsey-finite or Ramsey-infinite depending on the choice of G, H, k, and l with the general case unsettled . This disproves the conjecture given in [2] where it is suggested that the pair of graphs (L, M) is Ramsey-finite if and only if (1) either L or M is a matching, or (2) both L and M are star-forests of the type Sm U kS 1, m odd and k ~ 0 .