On Partitioning Colored Points

P. Kirchberger proved that, for a finite subset $X$ of ${\mathbb{R}}^d$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can be separated along the colors. In this paper, we show a more colorful theorem.