All finite sets are Ramsey in the maximum norm

TL;DR

This paper proves that for any finite metric space with at least two points, the minimum number of colors needed to avoid monochromatic copies in high-dimensional max-norm spaces grows exponentially with dimension.

Contribution

It establishes that all finite metric spaces are Ramsey in the maximum norm, providing exponential growth bounds for the chromatic number in high dimensions.

Findings

Chromatic number grows exponentially with dimension n

Explicit bounds provided for specific finite metric spaces

All finite metric spaces are Ramsey in the maximum norm

Abstract

For two metric spaces $\mathbb X$ and $\mathcal Y$, the chromatic number $\chi(\mathbb X;\mathcal Y)$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest $k$ such that there is a coloring of the points of $\mathbb X$ with $k$ colors and no monochromatic copy of $\mathcal Y$. In this paper, we show that for each finite metric space $\mathcal{M}$ that contains at least two points the value $\chi\left(\mathbb R^n_\infty; \mathcal M \right)$ grows exponentially with $n$. We also provide explicit lower and upper bounds for some special $\mathcal M$.