Generalised Ramsey numbers for two sets of cycles

Let C1 and C2 be two sets of cycles. We determine all generalised Ramsey numbers R(C1,C2) such that C1 or C2 contains a cycle of length at most 6. This generalises previous results of Erdős, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also provide a complete classification of most (C1,C2)-critical graphs such that C1 or C2 contains a cycle of length at most 5. For length 4, this is an easy extension of a recent result of Wu, Sun, and Radziszowski, in which |C1|=|C2|=1. For lengths 3 and 5, our results are new also in this special case.