Geodesic
On the colouring of spheres embedded in $\mathbb R^n$
Sbornik. Mathematics, Tome 202 (2011) no. 6, pp. 859-886 Cet article a éte moissonné depuis la source Math-Net.Ru

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The work concerns the well-known problem of identifying the chromatic number $\chi(\mathbb R^n)$ of the space $\mathbb R^n$, that is, finding the minimal number of colours required to colour all points of the space in such a way that any two points at distance one from each other have different colours. A new quantity generalising the chromatic number is introduced in the paper, namely, the speckledness of a subset in a fixed metric space. A series of lower bounds for the speckledness of spheres is derived. These bounds are used to obtain general results lifting lower bounds for the chromatic number of a space to higher dimensions, generalising the well-known ‘Moser-Raisky spindle’. As a corollary of these results, the best known lower bound for the chromatic number $\chi(\mathbb R^{12})\geqslant 27$ is obtained, and furthermore, the known bound $\chi(\mathbb R^4)\geqslant 7$ is reproved in several different ways. Bibliography: 23 titles.
Keywords: chromatic number, distance graph, speckledness of a set. 
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A. B. Kupavskii. On the colouring of spheres embedded in $\mathbb R^n$. Sbornik. Mathematics, Tome 202 (2011) no. 6, pp. 859-886. http://geodesic.mathdoc.fr/item/SM_2011_202_6_a3/

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