On the densities of cliques and independent sets in graphs

TL;DR

This paper determines the maximum number of red s-cliques in a red/blue edge coloring of a complete graph, given a fixed number of blue r-cliques, extending known results to all r and s using extremal combinatorics techniques.

Contribution

It generalizes the Kruskal-Katona theorem to arbitrary r and s by characterizing extremal colorings where edges form a clique.

Findings

Extremal coloring involves either all blue or all red edges forming a clique.

Provides a complete solution to the maximum red s-cliques problem given blue r-cliques.

Uses shifting technique and analytical methods from extremal set theory.

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.