On the densities of cliques and independent sets in graphs

Hao Huang, Nati Linial, Humberto Naves, Yuval Peled, Benny Sudakov

Published 2012-11-19, updated 2013-12-08Version 2

Let r, s >= 2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K_n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r=2 or s=2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.