Ramsey minimal graphs

As usual, for graphs G, G, and H, we write G ® (G, H) to mean that any red-blue colouring of the edges of G contains a red copy of G or a blue copy of H. A pair of graphs (G, H) is said to be Ramsey-infinite if there are infinitely many minimal graphs F for which we have G ® (G, H). Let l > 4 be an integer. We show that if H is a 2-connected graph that does not contain induced cycles of length at least l, then the pair (Ck,H) is Ramsey-infinite for any k > l, where Ck denotes the cycle of length k.

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