A generalization of the Bollobas set pairs inequality

The Bollobas set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for n >= k >= t >= 2, we consider a collection of k families A(i) : 1 <= i <= k where A(i) = {A(i,j) subset of [n] : j is an element of [n]} so that A(1),i(1) boolean AND ...boolean AND A(k,ik )not equal circle divide, if and only if there are at least t distinct indices i(1), i(2), ..., i(k). Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size beta(k,t)(n) of the families with ground set [n].