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2010 On $(2,3)$-agreeable box societies
Michael Abrahams, Meg Lippincott, Thierry Zell
Involve 3(1): 93-108 (2010). DOI: 10.2140/involve.2010.3.93


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 d2.


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Michael Abrahams. Meg Lippincott. Thierry Zell. "On $(2,3)$-agreeable box societies." Involve 3 (1) 93 - 108, 2010.


Received: 25 August 2009; Revised: 27 January 2010; Accepted: 9 February 2010; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1197.52006
MathSciNet: MR2672503
Digital Object Identifier: 10.2140/involve.2010.3.93

Primary: 52C45
Secondary: 91B12

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.3 • No. 1 • 2010
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