Coloring Planar Homothets and Three-Dimensional Hypergraphs

We prove that every finite set of homothetic copies of a given 1 compact and convex body in the plane can be colored with four colors 2 so that any point covered by at least two copies is covered by two copies 3 with distinct colors. This generalizes a previous result from Smorodinsky 4 (SIAM J. Disc. Math. 2007). Then we show that for any k ≥ 2, every 5 three-dimensional hypergraph can be colored with 6(k−1) colors so that 6 every hyperedge e contains min{|e|, k} vertices with mutually distinct 7 colors. This refines a previous result from Aloupis et al. (Disc. & Comp. 8 Geom. 2009). As corollaries, we improve on previous results for conflict9 free coloring, k-strong conflict-free coloring, and choosability. Proofs of 10 the upper bounds are constructive and yield simple, polynomial-time 11 algorithms. 12