Intersecting family

A family of sets is intersecting when every two of its members have nonempty intersection.

Two set families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting when $A\cap B\ne \varnothing$ for every $A\in \mathcal{A}$ and $B\in \mathcal{B}$.

The upward closure of $\mathcal{B}$ is $\overline{\mathcal{B}}=\{A:\text{some }B\in \mathcal{B}\text{ satisfies }B\subseteq A\}$.

For $0<p<1/2$ and $\epsilon >0$, every intersecting family is, up to ${\mu }_p$-measure $\epsilon$, contained in the upward closure of a set family generated by an intersecting family on a bounded set of coordinates.