  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 $\mathcal M$.
