Sperner theory
A localization inequality for set functions
Journal of Combinatorial Theory Series A
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Some inequalities for cross-unions of families of finite sets are proved that are related to the problem of minimizing the union-closure of a uniform family of given size. The cross-union of two families F and G of subsets of [n]={1,2,...,n} is the family F@?G={F@?G:F@?F,G@?G}. It is shown that |F@?G|/|F|=|G@?B"n|/2^n, where B"n denotes the power set of [n]. Besides, the problem of minimizing |F@?G| over all union-closed F and G generated by a given number r of singletons and a given number s(r2) of two-sets, respectively, is solved.