Given a metric space M that contains at least two points, the chromatic number χ (R n ∞ , M) is defined as the minimum number of colours needed to colour all points of an n−dimensional space R n ∞ with the max-norm such that no isometric copy of M is monochromatic. The last two authors have recently shown that the value χ (R n ∞ , M) grows exponentially for all finite M. In the present paper we refine this result by giving the exact value χ M such that χ (R n ∞ , M) = (χ M + o(1)) n for all 'one-dimensional' M and for some of their Cartesian products. We also study this question for infinite M. In particular, we construct an infinite M such that the chromatic number χ (R n ∞ , M) tends to infinity as n → ∞.

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 colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$ . In this article, 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 $\mathcal M$ .

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