The chromatic number of the space $(\mathbb R^n, l_1)$

The chromatic number of the space $(\mathbb R^n)$ with the $l_1$ metric $\| x\|=\sum_{i=1}^n{|x_i|}$ with $k$ forbidden distances is studied. It is defined as the minimum number of colours needed to colour all points in the space in such a way that no two points lying at a forbidden distance from each other in the $l_1$ metric have the same colour. The asymptotic growth of the chromatic numbers as $n\to\infty$ is estimated. The instrument of study is the linear algebra method, which reduces the problem of estimating chromatic numbers to another convex extremum problem. A numerical solution of this problem makes it possible to derive sharp estimates for the constants present in the bases of the lower asymptotic estimates for the chromatic numbers of multidimensional real spaces with several forbidden distances. The estimates obtained are optimal within the framework of the method proposed.
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