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 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 ARs(Kk) = Θ(k 6 / log2 k). 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