Rainbow Decompositions

A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson’s Theorem. For every integer k there exists an n0 = n0(k) so that for all n > n0, if n mod k(k − 1) ∈ {1, k}, then every properly edge-colored Kn contains (n 2 ) / (k 2 ) pairwise edge-disjoint rainbow copies of Kk. Our proof uses, as a main ingredient, a double application of the probabilistic method.