Rainbow triangles and the Caccetta-Häggkvist conjecture

Abstract: A famous conjecture of Caccetta and H\"aggkvist is that in a digraph on $n$ vertices and minimum out-degree at least $\frac{n}{r}$ 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 $\frac{n}{r}$, 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. For any fixed $k$ and large enough $n$, we determine the maximum number of edges in an $n$-vertex edge-coloured graph where all colour classes have size at most $k$ and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.