Indecomposable Coverings

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 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 ∈ D, all of whose points are of the same color. 1 Multiple arrangements: background and motivation The notion of multiple packings and coverings was introduced independently by Davenport and László Fejes Tóth. Given a system S of subsets of an underlying set X , we say that they form a k-fold packing (covering) if every point of X belongs to at most (at least) k members of S. A 1-fold packing (covering) is simply called a packing (covering). Clearly, the union of k packings (coverings) is always a k-fold packing (covering). Today there is a vast literature on this subject [FTG83], [FTK93]. Many results are concerned with the determination of the maximum density δ(C) of a k-fold packing (minimum density θ(C) of a k-fold covering) with congruent copies of a fixed convex body C. The same question was studied for multiple lattice packings (coverings), giving rise to the parameter δ L(C) (θ L(C)). Throughout this paper, it is always assumed that the geometric arrangements, packings, and coverings under consideration are locally finite, that ? 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. 2 János Pach, Gábor Tardos, and Géza Tóth is, any bounded region intersects only finitely many members of the arrrangement. Because of the strongly combinatorial flavor of the definitions, it is not surprising that combinatorial methods have played an important role in these investigations. For instance, Erdős and Rogers [ER62] used the “probabilistic method” to show that R can be covered with congruent copies (actually, with translates) of a convex body so that no point is covered more than e(d ln d + ln ln d + 4d) times (see [PA95], and [FuK05] for another combinatorial proof based on Lovász’ Local Lemma). Note that this easily implies that there exist positive constants θd, δd, depending only on d, such that k ≤ θ(C) ≤ kθ(C) ≤ kθd, kδd ≤ kδ(C) ≤ δ (C) ≤ k. Here δ(C) and θ(C) are shorthands for δ(C) and θ(C)). To establish almost tight density bounds, at least for lattice arrangements, it would be sufficient to show that any k-fold packing (covering) splits into roughly k packings (coverings), or into about k/l disjoint l-fold packings (coverings) for some l < k. The initial results were promising. Blundon [Bl57] and Heppes [He59] proved that for unit disks C = B, we have θ L(C) = 2θL(C), δ k L(C) = kδL(C) for k ≤ 4, and these results were extended to arbitrary centrally symmetric convex bodies in the plane by Dumir and Hans-Gill [DuH72] and by G. Fejes Tóth [FTG77], [FTG84]. In fact, there was a simple reason for this phenomenon: It turned out that every 3-fold lattice packing of the plane can be decomposed into 3 packings, and every 4-fold lattice packing into two 2-fold ones. This simple scheme breaks down for larger values of k. As k tends to infinity, Cohn [Co76] and Bolle [Bo89] proved that lim k→∞ θ L(C) k = lim k→∞ θ(C)