Non-bipartite pairs of 33-connected graphs are highly Ramsey-infinite

A pair of graphs (Hb,Hr) is highly Ramsey-infinite if there is some constant c such that for large enough n there are at least 2cn 2 non-isomorphic graphs on n or fewer vertices that are minimal with respect to the property that when their edges are coloured blue or red, there is necessarily a blue copy of Hb or a red copy of Hr. We show that a pair of 3-connected graphs is highly Ramsey-infinite if and only if at least one of the graphs in non-bipartite. Further we show that the pair (Hb,Hr) is highly Ramsey infinite for Hr an odd cycle of girth ` and Hb any graph with no induced cycle of length ` or longer. In showing the above results, we continue the theory of gadgets called senders and determiners that has been developed over many earlier papers on Ramsey-infinite graphs.