Geodesic
Indecomposable Coverings
Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 451-463

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DOI

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.
DOI : 10.4153/CMB-2009-048-x
Mots-clés : 52C15, 05C15 
Pach, János; Tardos, Gábor; Tóth, Géza. Indecomposable Coverings. Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 451-463. doi: 10.4153/CMB-2009-048-x
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     title = {Indecomposable {Coverings}},
     journal = {Canadian mathematical bulletin},
     pages = {451--463},
     year = {2009},
     volume = {52},
     number = {3},
     doi = {10.4153/CMB-2009-048-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-048-x/}
}
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