Finding Large p-Colored Diameter Two Subgraphs

Given a coloring of the edges of the complete graph K 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 k̿ and k\2hphk that there is always a p-colored diameter two subgraph of K_n containing at least ((k+p)n)/(2k)]\displaystyle{(k+p)n \over 2k} vertices and that this is best possible up to an additive constant l satisfying 0hl<k\2.