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

Let $S\subseteq {\mathbb{R}}^n$, and let $k\in \mathbb{N}$. Greenwell and Johnson define ${\widehat{\chi }}^{(k)}(S)$ 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 $C_1,\dots ,C_m$ such that no two points of $S$ which are a (Euclidean) distance $d_{ij}$ apart are both colored $C_j$, for all $1\le i\le k$ and $1\le j\le m$. If no such integer exists then we say that ${\widehat{\chi }}^{(k)}(S)=\mathrm{\infty }$. In this paper we show that ${\widehat{\chi }}^{(k)}(\mathbb{R})$ is finite for all $k$.