On (2, 3)-agreeable Box Societies

The notion of (k,m)-agreeable society was introduced by Deborah Berg et al.:a family of convex subsets of R 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 βn with non-empty intersection, where n is the size of the original family and β ∈ [0, 1] is an explicit constant depending only on k,m and d. The quantity β(k,m, d) is called the minimal agreement proportion for a (k,m)-agreeable family in R. If we only assume that the sets are convex, simple examples show that β = 0 for (k,m)-agreeable families in R 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 ≥ 2.