The (2,2) and (4,3) properties in families of fat sets in the plane [article]

Shiliang Gao, Shira Zerbib
<span title="2017-11-14">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A family of sets satisfies the (p,q) property if among every p members of it some q intersect. Given a number 0<r< 1, a set S⊂R^2 is called r-fat if there exists a point c∈ S such that B(c,r) ⊆ S⊆ B(c,1), where B(c,r)⊂R^2 is a disk of radius r with center-point c. We prove constant upper bounds C=C(r) on the piercing numbers in families of r-fat sets in R^2 that satisfy the (2,2) or the (4,3) properties. This extends results by Danzer and Karasev on the piercing numbers in intersecting families
more &raquo; ... of disks in the plane, as well as a result by Kynčl and Tancer on the piercing numbers in families of units disks in the plane satisfying the (4,3) property.
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