  Given a coloring of the edges of the complete graph on n vertices in k
colors, by considering the neighbors of an arbitrary vertex it follows that
there is a monochromatic diameter two subgraph on at least 1+(n-1)/k vertices.
We show that for $k \ge 3$ this is asymptotically best possible, and that for
k=2 there is always a monochromatic diameter two subgraph on at least $\lceil
{3 \over 4} n \rceil$ vertices, which again, is best possible.
