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A note on piercing discrete rectangles
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Paper
A note on piercing discrete rectangles
2026
Overview
In 2008, Halman proved a discrete Helly-type theorem for axis-parallel boxes in \\( R^d\\). Very recently, this result was extended to the \\((p,q)\\) setting with \\(p q d+1\\) by Edwards and Soberón, and subsequently to the case \\(p q 2\\) by Gangopadhyay, Polyanskii, and the author of this paper. In this paper, we obtain improved bounds for the \\((p,q)\\) problem in the case \\(q=2\\) and \\(d=2\\). More precisely, our main result asserts that for any integer \\(p 2\\), any set \\(P R^2\\), and any finite family \\( B\\) of axis-parallel rectangles in \\( R^2\\) such that every rectangle contains a point of \\(P\\), if among every \\(p\\) rectangles there exist two whose intersection contains a point of \\(P\\), then there exists a subset \\(S P\\) of size at most \\(O\\!( (p p)^2 )\\) such that every rectangle contains a point of \\(S\\). Moreover, when \\(p=2\\), the size of \\(S\\) can be bounded by \\(4\\).
Publisher
Cornell University Library, arXiv.org
Subject
