A pair of graphs is highly Ramsey-infinite if there is some constant such that for large enough there are at least non-isomorphic graphs on 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 or a red copy of .
We show that a pair of -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 is highly Ramsey infinite for an odd cycle of girth and 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.