On the upper chromatic numbers of the reals Original Research Article

Let S be a metric space and let k be a positive integer. Define χ̂(k)(S) to be the smallest positive integer m such that for every k×m array D=(Dij) of positive real numbers, S can be colored with the colors 1,2,…,m such that no two points of distance Dij are both colored j. We improve the best upper bound known on χ̂(k)(R) from 32kk! to ⌈4ek⌉, where e is the base of the natural logarithm. We prove a conjecture of Abrams (Discrete Math. 169 (1997) 157–162) that χ̂(k)(Z)=χ̂(k)(R) for all k∈N, extend this result to higher dimensions under the l1 and l∞ norms, and prove that the upper chromatic numbers are finite for these spaces. We also introduce a new related chromatic quantity of a graph G, the chromatic capacity, χcap(G).