On (2,3)-agreeable Box Societies

Details On (2,3)-agreeable Box Societies On (2,3)-agreeable Box Societies

2009-08-25

arxiv.org/abs/0908.3692v1

Mathematics

Combinatorics

15 pages, 10 figures

Scientific paper

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

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