The $k^{\text th}$ Upper Chromatic Number of the Line

Let $S \subseteq \mathbb{R}^n$, and let $k\in\mathbb{N}$. Greenwell and Johnson define ${\hat\chi\ }^{(k)}(S)$ to be the smallest integer $m$ (if such an integer exists) such that for every $k\times m$ array $D=(d_{ij})$ of positive real numbers, $S$ can be colored with the colors $C_1,\ldots,C_m$ such that no two points of $S$ which are a (Euclidean) distance $d_{ij}$ apart are both colored $C_j$, for all $1\leq i \leq k$ and $1\leq j \leq m$. If no such integer exists then we say that ${\hat\chi\ }^{(k)}(S)=\infty$. In this paper we show that ${\hat\chi\ }^{(k)}(\mathbb{R})$ is finite for all $k$.