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/

[1] K. Borsuk, “Drei Sätze über die $n$-dimensionale euklidische Sphäre”, Fundam. Math., 20 (1933), 177–190 | Zbl

[2] J. Kahn, G. Kalai, “A counterexample to Borsuk's conjecture”, Bull. Amer. Math. Soc. (N.S.), 29:1 (1993), 60–62 | DOI | MR | Zbl

[3] H. Hadwiger, “Ein Überdeckungssatz für den Euklidischen Raum”, Portugaliae Math., 4 (1944), 140–144 | MR | Zbl

[4] A. Soifer, Mathematical Coloring Book, Center for Excellence in Math. Education, Colorado Springs, 1997

[5] V. G. Boltyanskii, I. Ts. Gokhberg, Teoremy i zadachi kombinatornoi geometrii, Nauka, M., 1965 | MR

[6] V. G. Boltyanski, H. Martini, P. S. Soltan, Excursions into Combinatorial Geometry, Universitext, Springer-Verlag, Berlin, 1997 | MR | Zbl

[7] A. M. Raigorodskii, “Problema Borsuka i khromaticheskie chisla nekotorykh metricheskikh prostranstv”, UMN, 56:1 (2001), 107–146 | MR | Zbl

[8] A. Soifer, “Chromatic number of the plane: a historical essay”, Geombinatorics, 1:3 (1991), 13–15 | MR | Zbl

[9] A. M. Raigorodskii, “The Borsuk partition problem: the seventieth anniversary”, Math. Intelligencer, 26:3 (2004), 4–12 | DOI | MR | Zbl

[10] A. M. Raigorodskii, “O nizhnikh otsenkakh dlya chisel Borsuka i Khadvigera”, UMN, 59:3 (2004), 177–178 | MR | Zbl

[11] A. M. Raigorodskii, “O svyazi mezhdu problemami Borsuka i Erdesha–Khadvigera”, UMN, 60:4 (2005), 219–220 | MR | Zbl

[12] A. M. Raigorodskii, “O chislakh Borsuka i Erdesha–Khadvigera”, Matem. zametki, 79:6 (2006), 913–924 | MR | Zbl

[13] P. Frankl, R. Wilson, “Intersection theorems with geometric consequences”, Combinatorica, 1:4 (1981), 357–368 | DOI | MR | Zbl

[14] I. M. Shitova, “O khromaticheskikh chislakh metricheskikh prostranstv s dvumya zapreschennymi rasstoyaniyami”, Dokl. RAN, 413:2 (2007), 178–180

[15] K. Prakhar, Raspredelenie prostykh chisel, Mir, M., 1967 | MR | Zbl

[16] A. M. Raigorodskii, “Khromaticheskie chisla distantsionnykh grafov”, Chebyshevskii sb., 6:3 (2005), 159–170 | MR | Zbl

[17] F. Kharari, Teoriya grafov, Mir, M., 1973 | MR | Zbl

[18] A. M. Raigorodskii, Lineino-algebraicheskii metod v kombinatorike, MTsNMO, M., 2007