On Partitioning Colored Points

TL;DR

This paper extends Kirchberger's theorem to a more colorful setting, providing new insights into partitioning colored points in Euclidean space.

Contribution

It introduces a generalized colorful theorem for separating colored points, expanding upon Kirchberger's classical result.

Findings

Generalized Kirchberger's theorem for colored points

Proved that certain local separations imply global separation in colored point sets

Enhanced understanding of partitioning in multi-colored Euclidean point configurations

Abstract

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.