Rainbow triangles and the Caccetta-Haggkvist conjecture

A famous conjecture of Caccetta and Haggkvist is that in a digraph on n vertices and minimum outdegree at least n/r there is a directed cycle of length r or less. We consider the following generalization: in an undirected graph on n vertices, any collection of n disjoint sets of edges, each of size at least n/r, has a rainbow cycle of length r or less. We focus on the case r = 3 and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. In our main result, whenever n is larger than a suitable polynomial in k, we determine the maximum number of edges in an n-vertex edge-colored graph where all color classes have size at most k and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.