Finding large p-colored diameter two subgraphs

Given a coloring of the edges of the complete graph K-n on n vertices in k colors, a p-colored subgraph of K-n is any subgraph whose edges only use colors from some p element set. We show for k greater than or equal to 1 and k/2 less than or equal to p less than or equal to k that there is always a p-colored diameter two subgraph of K-n containing at least (k + p)n/2k vertices and that this is best possible up to an additive constant Isatisfying 0 less than or equal to l < k/2.