Coloring d-Embeddable k-Uniform Hypergraphs

This paper extends the scenario of the Four Color Theorem in the following way. LetHd,k be the set of all k-uniform hypergraphs that can be (linearly) embedded into Rd . We investigate lower and upper bounds on the maximum (weak and strong) chromatic number of hypergraphs in Hd,k. For example, we can prove that for d ≥ 3 there are hypergraphs inH2d−3,d on n vertices whose weak chromatic number is Ω(log n/ log log n), whereas the weak chromatic number for n-vertex hypergraphs inHd,d is bounded by O (n(d−2)/(d−1)) for d ≥ 3.