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Indecomposable Coverings

Published online by Cambridge University Press:  20 November 2018

János Pach
Affiliation:
City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. e-mail: pach@cims.nyu.edu
Gábor Tardos
Affiliation:
School of Computer Science, Simon Fraser University, Burnaby, BC, V5A 1S6 e-mail: tardos@cs.sfu.ca
Géza Tóth
Affiliation:
Rényi Institute, Hungarian Academy of Sciences, P.O.B. 127 Budapest, 1364, Hungary e-mail: geza@renyi.hu
Corresponding
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Abstract

We prove that for every $k\,>\,1$ , there exist $k$ -fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct for every $k\,>\,1$ a set of points $P$ and a family of disks $D$ in the plane, each containing at least $k$ elements of $P$ , such that, no matter how we color the points of $P$ with two colors, there exists a disk $D\,\in \,D$ all of whose points are of the same color.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2009

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