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2013 | 33 | 1 | 25-31
Tytuł artykułu

Coloring Some Finite Sets in ℝn

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EN
Abstrakty
EN
This note relates to bounds on the chromatic number χ(ℝn) of the Euclidean space, which is the minimum number of colors needed to color all the points in ℝn so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in ℝn was introduced showing that . For many years, this bound has been remaining the best known bound for the chromatic numbers of some lowdimensional spaces. Here we prove that and find an exact formula for the chromatic number in the case of n = 2k and n = 2k − 1.
Wydawca
Rocznik
Tom
33
Numer
1
Strony
25-31
Opis fizyczny
Daty
wydano
2013-03-01
online
2013-04-13
Twórcy
  • Department of Mathematics, University of Illinois, Urbana, IL 61801, USA, jobal@math.uiuc.edu
  • Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA, Sobolev Institute of Mathematics, Novosibirsk, Russia, kostochk@math.uiuc.edu
  • Department of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119991, Russia Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Russia, mraigor@yandex.ru
Bibliografia
  • [1] N.G. de Bruijn and P. Erdős, A colour problem for infinite graphs and a problem in the theory of relations, Proc. Koninkl. Nederl. Acad. Wet. (A) 54 (1951) 371-373.
  • [2] P. Frankl and R. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981) 357-368. doi:10.1007/BF02579457[Crossref]
  • [3] A.B. Kupavskiy, On coloring spheres embedded into Rn, Sb. Math. 202(6) (2011) 83-110.
  • [4] A.B. Kupavskiy and A.M. Raigorodskii, On the chromatic number of R9, J. Math. Sci. 163(6) (2009) 720-731. doi:10.1007/s10958-009-9708-4[Crossref]
  • [5] D.G. Larman, A note on the realization of distances within sets in Euclidean space, Comment. Math. Helv. 53 (1978) 529-535. doi:10.1007/BF02566096[Crossref]
  • [6] D.G. Larman and C.A. Rogers, The realization of distances within sets in Euclidean space, Mathematika 19 (1972) 1-24. doi:10.1112/S0025579300004903[Crossref]
  • [7] N. Pippenger and J. Spencer, Asymptotic behavior of the chromatic index for hypergraphs, J. Combin. Theory (A) 51 (1989) 24-42. doi:10.1016/0097-3165(89)90074-5[Crossref]
  • [8] A.M. Raigorodskii, On the chromatic number of a space, Russian Math. Surveys 55 (2000) N2, 351-352. doi:10.1070/RM2000v055n02ABEH000281[Crossref]
  • [9] A.M. Raigorodskii, The problems of Borsuk and Grünbaum on lattice polytopes, Izv. Math. 69(3) (2005) 81-108. English transl. Izv. Math. 69(3) (2005) 513-537. doi:10.1070/IM2005v069n03ABEH000537[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1641
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