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.
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Research supported in part by NSF grant DMS-1128155.
Research supported in part by the Israel Science Foundation and by a USA-Israel BSF grant.
Research supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant.
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Huang, H., Linial, N., Naves, H. et al. On the densities of cliques and independent sets in graphs. Combinatorica 36, 493–512 (2016). https://doi.org/10.1007/s00493-015-3051-9
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DOI: https://doi.org/10.1007/s00493-015-3051-9