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 c(k)n(k), where c(k) is an element of (0, 1) is the unique root of the equation z(k) = (1 - z)(k) + kz(1 - z)(k-1). On the other hand, we construct a coloring in which there are at least c(k)n(k) - O(n(k-1)) red k-cliques and at least the same number of blue k-cliques. (C) 2013 Elsevier Inc. All rights reserved.