A generalization of the Bollobás set pairs inequality (Q2040005)

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