Two-colorings with many monochromatic cliques in both colors

Color the edges of the n-vertex complete graph in red and blue, and suppose that red k-cliques are fewer than blue k-cliques. We show that the number of red k-cliques is always less than cknk, where ck∈(0,1) is the unique root of the equation zk=(1−z)k+kz(1−z)k−1. On the other hand, we construct a coloring in which there are at least cknk−O(nk−1) red k-cliques and at least the same number of blue k-cliques.