Geodesic
On a Series of Problems Related to the Borsuk and Nelson--Erd\H os--Hadwiger Problems
Matematičeskie zametki, Tome 84 (2008) no. 2, pp. 254-272

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In the present paper, a series of problems connecting the Borsuk and Nelson–Erdős–Hadwiger classical problems in combinatorial geometry is considered. The problem has to do with finding the number $\chi(n,a,d)$ equal to the minimal number of colors needed to color an arbitrary set of diameter $d$ in $n$-dimensional Euclidean space in such a way that the distance between points of the same color cannot be equal to $a$. Some new lower bounds for the quantity $\chi(n,a,d)$ are obtained.
Keywords: Borsuk problem, Nelson–Erdős–Hadwiger problem, chromatic number, Stirling formula, infinite graph, Euclidean space, distribution of primes. 
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A. M. Raigorodskii; M. M. Kityaev. On a Series of Problems Related to the Borsuk and Nelson--Erd\H os--Hadwiger Problems. Matematičeskie zametki, Tome 84 (2008) no. 2, pp. 254-272. http://geodesic.mathdoc.fr/item/MZM_2008_84_2_a7/