Rainbow transitive triangles in arc-colored digraphs

A subdigraph of an arc-colored digraph is rainbow if its all arcs have distinct colors. For two digraphs D and H, let rb(D,H) be the minimum integer such that every arc-colored digraph D^C with c(D)≥rb(D,H) contains a rainbow copy of H, where c(D) is the number of colors of D^C. Let K_n↔ be the digraph obtained from the complete graph K_n by replacing each edge uv with a pair of symmetric arcs (u,v) and (v,u), and let T₃⃗ be the transitive triangle. In this paper we determine rb(K_n↔,T₃⃗) and characterize the corresponding extremal arc-colorings of K_n↔. Further, we prove that an arc-colored digraph D^C on n vertices contains a rainbow T₃⃗ if a(D)+c(D)≥a(K_n↔)+rb(K_n↔,T₃⃗). Moreover, if a(D)+c(D)=a(K_n↔)+rb(K_n↔,T₃⃗)−1 and D^C contains no rainbow T₃⃗’s, then D≅K_n↔.