On $(2,3)$-agreeable box societies

Abstract

The notion of a $\left(k,m\right)$-agreeable society was introduced by Berg, Norine, Su, Thomas and Wollan: a family of convex subsets of ${ℝ}^d$ is called $\left(k,m\right)$-agreeable if any subfamily of size $m$ contains at least one nonempty $k$-fold intersection. In that paper, the $\left(k,m\right)$-agreeability of a convex family was shown to imply the existence of a subfamily of size $\beta n$ with a nonempty intersection, where $n$ is the size of the original family and $\beta \in \left[0,1\right]$ is an explicit constant depending only on $k$, $m$ and $d$. The quantity $\beta \left(k,m,d\right)$ is called the minimal agreement proportion for a $\left(k,m\right)$-agreeable family in ${ℝ}^d$.

If we assume only that the sets are convex, simple examples show that $\beta =0$ for $\left(k,m\right)$-agreeable families in ${ℝ}^d$ where $k<d$. In this paper, we introduce new techniques to find positive lower bounds when restricting our attention to families of $d$-boxes, that is, cuboids with sides parallel to the coordinates hyperplanes. We derive explicit formulas for the first nontrivial case: $\left(2,3\right)$-agreeable families of $d$-boxes with $d\ge 2$.