Directed Graphs Without Rainbow Triangles

One of the most fundamental results in graph theory is Mantel's theorem which determines the maximum number of edges in a triangle-free graph of order $n$. Recently, a colorful variant of this problem has been solved. In this variant we consider $c$ graphs on a common vertex set, think of each graph as edges in a distinct color, and want to determine the smallest number of edges in each color which guarantees the existence of a rainbow triangle. Here, we solve the analogous problem for directed graphs without rainbow triangles, either directed or transitive, for any number of colors. The constructions and proofs essentially differ for $c=3$ and $c\ge 4$ and the type of the forbidden triangle. Additionally, we also solve the analogous problem in the setting of oriented graphs.