Geodesic
A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics $\ell_1$ and $\ell_2$
Matematičeskie zametki, Tome 105 (2019) no. 2, pp. 187-213

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A particular class of estimates related to the Nelson–Erdős–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric $\ell_1$, certain series of distance graphs of small dimensions are considered. Independence numbers of such graphs are estimated by using the linear-algebraic method and combinatorial observations. This makes it possible to obtain certain lower bounds for the chromatic numbers of the spaces mentioned above and, for each case, specify a series of graphs leading to the strongest results.
Keywords: chromatic number, chromatic number of a metric space, independence number, linear-algebraic method, distance graph. 
@article{MZM_2019_105_2_a2,
     author = {L. I. Bogolubsky and A. M. Raigorodskii},
     title = {A {Remark} on {Lower} {Bounds} for the {Chromatic} {Numbers} of {Spaces} of {Small} {Dimension} with {Metrics} $\ell_1$ and $\ell_2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {187--213},
     publisher = {mathdoc},
     volume = {105},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_105_2_a2/}
}
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L. I. Bogolubsky; A. M. Raigorodskii. A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics $\ell_1$ and $\ell_2$. Matematičeskie zametki, Tome 105 (2019) no. 2, pp. 187-213. http://geodesic.mathdoc.fr/item/MZM_2019_105_2_a2/