All finite sets are Ramsey in the maximum norm
Résumé
For two metric spaces $\mathcal X$ and $\mathcal Y$, the chromatic number $\chi(\mathcal X;\mathcal Y)$ of $\mathcal X$ with forbidden $\mathcal Y$ is the smallest $k$ such that there is a coloring of the points of $\mathcal X$ with no monochromatic copy of $\mathcal Y$. In this paper, we show that for each finite metric space $\mathcal{M}$ 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 $\mathcal M$.