A generalization of the Bollobás set pairs inequality

Abstract

The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $n\ge k\ge t\ge 2$, we consider a collection of $k$ families ${\mathcal{A}}_i:1\le i\le k$ where ${\mathcal{A}}_i=\{A_{i,j}\subset [n]:j\in [n]\}$ so that $A_{1,i_1}\cap \cdots \cap A_{k,i_k}\ne \varnothing$ 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 $[n]$.

Revised writing from previous version