Abstract
We prove that for every k > 1, there exist k-fold coverings of the plane (1) with strips, (2) with axis-parallel rectangles, and (3) 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 \(\cal D\) 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 \(D\in{\cal D}\), all of whose points are of the same color.
János Pach has been supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, Hungarian Research Foundation OTKA, and BSF. Gábor Tardos has been supported by OTKA T-046234, AT-048826, and NK-62321. Géza Tóth has been supported by OTKA T-038397.
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Pach, J., Tardos, G., Tóth, G. (2007). Indecomposable Coverings. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds) Discrete Geometry, Combinatorics and Graph Theory. CJCDGCGT 2005. Lecture Notes in Computer Science, vol 4381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70666-3_15
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DOI: https://doi.org/10.1007/978-3-540-70666-3_15
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