Indecomposable Coverings

http://dx.doi.org/10.4153/CMB-2009-048-x
Canad. Math. Bull. 52(2009), 451-463

  Published:2009-09-01
 Printed: Sep 2009

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 $\cal D$ 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.