Directed Graphs Without Rainbow Triangles

ABSTRACT 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 . Recently, a colorful variant of this problem has been solved. In this variant we consider 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 and and the type of the forbidden triangle. Additionally, we also solve the analogous problem in the setting of oriented graphs.