Coloring planar homothets and three-dimensional hypergraphs

We prove that every finite set of homothetic copies of a given convex body in the plane can be colored with four colors so that any point covered by at least two copies is covered by two copies with distinct colors. This generalizes a previous result from Smorodinsky (SIAM J. Disc. Math. 2007). Then we show that for any k≥2, every three-dimensional hypergraph can be colored with 6(k-1) colors so that every hyperedge e contains min{|e|,k} vertices with mutually distinct colors. This refines a previous result from Aloupis et al. (Disc. & Comp. Geom. 2009). As corollaries, we improve on previous results for conflict-free coloring, k-strong conflict-free coloring, and choosability. Proofs of the upper bounds are constructive and yield simple, polynomial-time algorithms.