Size and Degree Anti-Ramsey Numbers

A copy of a graph H in an edge colored graph G is called rainbow if all edges of H have distinct colors. The size anti-Ramsey number of H, denoted by $$AR_s(H)$$ARs(H), is the smallest number of edges in a graph G such that any of its proper edge-colorings contains a rainbow copy of H. We show that $$AR_s(K_k) = \varTheta (k^6 / \log ^2 k)$$ARs(Kk)=Θ(k6/log2k). This settles a problem of Axenovich, Knauer, Stumpp and Ueckerdt. The proof is probabilistic and suggests the investigation of a related notion, which we call the degree anti-Ramsey number of a graph.