Vol. 3, No. 1, 2010

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

Michael Abrahams, Meg Lippincott and Thierry Zell

Vol. 3 (2010), No. 1, 93–108
Abstract

The notion of a (k,m)-agreeable society was introduced by Berg, Norine, Su, Thomas and Wollan: a family of convex subsets of d is called (k,m)-agreeable if any subfamily of size m contains at least one nonempty 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 a nonempty 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 d.

If we assume only that the sets are convex, simple examples show that β = 0 for (k,m)-agreeable families in 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: (2,3)-agreeable families of d-boxes with d 2.

Keywords
boxicity, arrangements of boxes, agreement proportion, voting, Helly's theorem
Mathematical Subject Classification 2000
Primary: 52C45
Secondary: 91B12
Milestones
Received: 25 August 2009
Revised: 27 January 2010
Accepted: 9 February 2010
Published: 20 April 2010

Communicated by Arthur T. Benjamin
Authors
Michael Abrahams
Vassar College
Poughkeepsie, NY 12604
United States
Meg Lippincott
Vassar College
Poughkeepsie, NY 12604
United States
Thierry Zell
The Donald and Helen Schort School of Mathematics and Computing Sciences
Lenoir-Rhyne University
Hickory NC 28603
United States
http://mat.lr.edu/faculty/zell