Inequalities for cross-unions of collections of finite sets

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 > ( r 2 ) of two-sets, respectively, is solved.