Variations on the Bollobás set-pair theorem

Let X be an n-element set. A set-pair system P={(Ai,Bi)}1≤i≤m is a collection of pairs of disjoint subsets of X. It is called skew Bollobás system if Ai∩Bj≠0̸ for all 1≤i<j≤m. The best possible inequality ∑i=1m1|Ai|+|Bi||Ai|≤n+1. is established along with some more results of similar flavor.