Max-norm Ramsey Theory

Given a metric space $\mathcal{M}$ that contains at least two points, the chromatic number $\chi ({\mathbb{R}}_{\mathrm{\infty }}^n,\mathcal{M})$ is defined as the minimum number of colours needed to colour all points of an $n$-dimensional space ${\mathbb{R}}_{\mathrm{\infty }}^n$ with the max-norm such that no isometric copy of $\mathcal{M}$ is monochromatic. The last two authors have recently shown that the value $\chi\left($\mathbb{R}$^n_{\infty}, $\mathcal{M}$ \right)$growsexponentiallyforallfinite$$\mathcal{M}$$. In the present paper we refine this result by giving the exact value ${\chi }_{\mathcal{M}}$ such that $\chi\left($\mathbb{R}$^n_{\infty}, $\mathcal{M}$ \right) = (\chi_{$\mathcal{M}$}+o(1))^n$foral{l}^{\mathrm{\prime }}one-dimensiona{l}^{\mathrm{\prime }}$$\mathcal{M}$$ and for some of their Cartesian products. We also study this question for infinite $\mathcal{M}$. In particular, we construct an infinite $\mathcal{M}$ such that the chromatic number $\chi ({\mathbb{R}}_{\mathrm{\infty }}^n,\mathcal{M})$ tends to infinity as $n\to \mathrm{\infty }$.