The kth upper chromatic number of the line

Let S ⊆ Rn, and let k ϵ N. Greenwell and Johnson [3] define <χ(k) (S) to be the smallest integer m (if such an integer exists) such that for every k × m array D = (dij) of positive real numbers, S can be colored with the colors C1, …, Cm such that no two points of S which are a (Euclidean) distance dij apart are both colored Cj, for all 1 ⩽i⩽k and 1⩽j⩽m. If no such integer exists then we say that <χ(k)(S) = ∞. In this paper we show that <χ(k) (R) is finite for all k.