Polychromatic Colorings of Unions of Geometric Hypergraphs

A polychromatic k-coloring of a hypergraph assigns to each vertex one of k colors in such a way that every hyperedge contains all the colors. A range capturing hypergraph is an m-uniform hypergraph whose vertices are points in the plane and whose hyperedges are those m-subsets of points that can be separated by some geometric object of a particular type, such as axis-aligned rectangles, from the remaining points. Polychromatic k-colorings of m-uniform range capturing hypergraphs are motivated by the study of weak epsilon-nets and cover decomposability problems. We show that the hypergraphs in which each hyperedge is determined by a bottomless rectangle or by a horizontal strip in general do not allow for polychromatic colorings. This strengthens the corresponding result of Chen, Pach, Szegedy, and Tardos [Random Struct. Algorithms, 34:11-23, 2009] for axis-aligned rectangles, and gives the first explicit (not randomized) construction of non-2-colorable hypergraphs defined by axis-aligned rectangles of arbitrarily large uniformity. In general we consider unions of range capturing hypergraphs, each defined by a type of unbounded axis-aligned rectangles. For each combination of types, we show that the unions of such hypergraphs either admit polychromatic k-colorings for m = O(k), m = O (k log k), m = O(k(8.75)), or do not admit in general polychromatic 2-colorings for any m.