Generalized rainbow Turán numbers of odd cycles

Abstract

Given graphs F and H, the generalized rainbow Turán numberex(n,F,rainbow-H) is the maximum number of copies of F in an n-vertex graph with a proper edge-coloring that contains no rainbow copy of H. B. Janzer determined the order of magnitude of ex(n,Cs,rainbow-Ct) for all s≥4 and t≥3, and a recent result of O. Janzer implied that ex(n,C3,rainbow-C2k)=O(n1+1/k). We prove the corresponding upper bound for the remaining cases, showing that ex(n,C3,rainbow-C2k+1)=O(n1+1/k). This matches the known lower bound for k even and is conjectured to be tight for k odd.