For graphs G,H, we write Grb⟶H if for every proper edge-coloring of G there is a rainbow copy of H, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last author proved that the threshold for G(n,p)rb⟶H is at most n−1/m2(H). Previous results have matched the lower bound for this anti-Ramsey threshold for cycles and complete graphs with at least 5 vertices. Kohayakawa, Konstadinidis and the last author also presented an infinite family of graphs H for which the anti-Ramsey threshold is asymptotically smaller than n−1/m2(H). In this paper, we devise a framework that provides a richer family of such graphs.
