Short rainbow cycles for families of matchings and triangles

Abstract A generalization of the famous Caccetta–Häggkvist conjecture, suggested by Aharoni, is that any family of sets of edges in , each of size , has a rainbow cycle of length at most . In works by the author with Aharoni and by the author with Aharoni, Berger, Chudnovsky, and Zerbib, it was shown that asymptotically this can be improved to if all sets are matchings of size 2, or all are triangles. We show that the same is true in the mixed case, that is, if each is either a matching of size 2 or a triangle. We also study the case that each is a matching of size 2 or a single edge, or each is a triangle or a single edge, and in each of these cases we determine the threshold proportion between the types, beyond which the rainbow girth goes from linear to logarithmic.