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On the densities of cliques and independent sets in graphs. (English) Zbl 1399.05125
Summary: Let $$r$$, $$s\geq 2$$ 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.

##### MSC:
 05C35 Extremal problems in graph theory 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05D05 Extremal set theory
##### Keywords:
coloring; $$K_n$$ graph
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##### References:
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