Chromatic Number with Several Forbidden Distances in the Space with the ℓq-Metric

Abstract

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