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

A pair of graphs ( H b , H r ) is highly Ramsey-infinite if there is some constant c such that for large enough n there are at least 2 c n 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 H b or a red copy of H r . w 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 ( H b , H r ) is highly Ramsey infinite for H r an odd cycle of girth ℓ and H b any graph with no induced cycle of length ℓ or longer. wing 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.