Chromatic number with several forbidden distances in the space with the $\ell_q$-metric

We study the chromatic number $\overline\chi(X; \rho; k)$ of a metric space $X$ with a metric $\rho$ and $k$ forbidden distances. We obtain an estimate of the form $\overline\chi({R}^n; \rho; k) \geq (Bk)^{Cn}$ for cases where the metric $\rho$ on the set $\mathbb{R}^n$ is generated by the $\ell_q$-norm.