On (2,3)-agreeable Box Societies

TL;DR

This paper investigates the agreement properties of families of $d$-boxes in Euclidean space, establishing explicit lower bounds for the proportion of boxes with a common intersection in the case of $(2,3)$-agreeability.

Contribution

It introduces new techniques to derive positive lower bounds for agreement proportions specifically for families of $d$-boxes, focusing on the $(2,3)$-agreeable case.

Findings

Derived explicit formulas for the agreement proportion in $(2,3)$-agreeable $d$-box families.

Showed that positive agreement bounds exist for $d$-boxes when $k=d$, unlike the general convex case.

Extended understanding of intersection properties in geometric families of boxes.

Abstract

The notion of $(k,m)$-agreeable society was introduced by Deborah Berg et al.: a family of convex subsets of $\R^d$ is called $(k,m)$-agreeable if any subfamily of size $m$ contains at least one non-empty $k$-fold intersection. In that paper, the $(k,m)$-agreeability of a convex family was shown to imply the existence of a subfamily of size $\beta n$ with non-empty intersection, where $n$ is the size of the original family and $\beta\in[0,1]$ is an explicit constant depending only on $k,m$ and $d$. The quantity $\beta(k,m,d)$ is called the minimal \emph{agreement proportion} for a $(k,m)$-agreeable family in $\R^d$. If we only assume that the sets are convex, simple examples show that $\beta=0$ for $(k,m)$-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, i.e. cuboids with sides parallel to the coordinates hyperplanes. We derive explicit formulas for the first non-trivial case: the case of $(2,3)$-agreeable families of $d$-boxes with $d\geq 2$.