SUMMARY
The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for n?n \geqslant k \geqslant t \geqslant 2, we consider a collection of kk families \mathcal{A}_i: 1 \leq i \leqslant k\mathcal{A}_i: 1 \leq i \leqslant k where \mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \} so that A_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothingA_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothing if and only if there are at least tt distinct indices i_1,i_2,\dots,i_ki_1,i_2,\dots,i_k. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size \beta_{k,t}(n)\beta_{k,t}(n) of the families with ground set [n][n].