Indecomposable Coverings

We prove that for every k > 1, there exist k-fold coverings of the plane (i) with strips, 00 with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed in to 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 is an element of D all of whose points are of the same color.