On the densities of cliques and independent sets in graphs
Combinatorics
2013-12-10 v2
Abstract
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.
Keywords
Cite
@article{arxiv.1211.4532,
title = {On the densities of cliques and independent sets in graphs},
author = {Hao Huang and Nati Linial and Humberto Naves and Yuval Peled and Benny Sudakov},
journal= {arXiv preprint arXiv:1211.4532},
year = {2013}
}