Geodesic
On the Chromatic Numbers of Rational Spaces
Matematičeskie zametki, Tome 103 (2018) no. 1, pp. 120-128.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper, we consider the notion of affine chromatic number of a rational space introduced by E. I. Ponomarenko and A. M. Raigorodskii and describe its relationship with chromatic numbers of rational spaces.
Keywords: chromatic number, rational space, unit distance graph. 
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A. Sokolov. On the Chromatic Numbers of Rational Spaces. Matematičeskie zametki, Tome 103 (2018) no. 1, pp. 120-128. http://geodesic.mathdoc.fr/item/MZM_2018_103_1_a10/

[1] A. Soifer, The Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of its Creators, Springer, New York, 2009 | MR | Zbl

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

[3] A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters”, Thirty Essays on Geometric Graph Theory, Springer, New York, 2013, 429–460 | MR | Zbl

[4] O. Nechushtan, “Note on the space chromatic number”, Discrete Math., 256:1-2 (2002), 499–507 | DOI | MR | Zbl

[5] D. Coulson, “A 15-Coloring of 3-Space omitting distance one”, Discrete Math., 256:1-2 (2002), 83–90 | DOI | MR | Zbl

[6] A. M. Raigorodskii, “O khromaticheskom chisle prostranstva”, UMN, 55:2 (332) (2000), 147–148 | DOI | MR | Zbl

[7] D. G. Larman, C. A. Rogers, “The realization of distances within sets in Euclidean space”, Mathematika, 19 (1972), 1–24 | DOI | MR | Zbl

[8] M. Benda, M. Perles, “Colorings of metric spaces”, Geombinatorics, 9:3 (2000), 113–126 | MR | Zbl

[9] D. R. Woodall, “Distances realized by sets covering the plane”, J. Combinatorial Theory Ser. A, 14 (1973), 187–200 | DOI | MR | Zbl

[10] E. I. Ponomarenko, A. M. Raigorodskii, “Novye otsenki v zadache o chisle reber gipergrafa s zapretami na peresecheniya”, Probl. peredachi inform., 49:4 (2013), 98–104 | MR | Zbl

[11] E. I. Ponomarenko, A. M. Raigorodskii, “Novaya nizhnyaya otsenka khromaticheskogo chisla ratsionalnogo prostranstva s odnim i dvumya zapreschennymi rasstoyaniyami”, Matem. zametki, 97:2 (2015), 255–261 | DOI | MR | Zbl

[12] P. Johnson, A. Schneider, M. Tiemeyer, “$B_1(\mathbb Q^3)=4$”, Geombinatorics, 16:4 (2007), 356–362 | MR

[13] E. I. Ponomarenko, A. M. Raigorodskii, “O khromaticheskom chisle prostranstva $\mathbb Q^n$”, Tr. MFTI, 4:1 (2012), 127–130

[14] P. Erdős, “Graph theory and probability”, Canad. J. Math., 11 (1959), 34–38 | DOI | MR | Zbl

[15] M. Axenovich, J. Choi, M. Lastrina, T. McKay, J. Smith, B. Stanton, “On the Chromatic Number of Subsets of the Euclidean Plane”, Graphs Combin., 30:1 (2014), 71–81 | DOI | MR | Zbl

[16] K. Fischer, “Additive $K$-colorable extensions of the rational plane”, Discrete Math., 82:2 (1990), 181–195 | DOI | MR | Zbl

[17] P. D. Johnson, Jr., “Two-colorings of real quadratic extensions of $\mathbb Q^2$ that forbid many distances”, Congr. Numer., 60 (1987), 51–58 | MR | Zbl

[18] A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters”, Discrete Geometry and Algebraic Combinatorics, Contemp. Math., 625, Amer. Math. Soc., Providence, RI, 2014, 93–109 | MR | Zbl

[19] A. M. Raigorodskii, “Problema Borsuka i khromaticheskie chisla nekotorykh metricheskikh prostranstv”, UMN, 56:1 (337) (2001), 107–146 | DOI | MR | Zbl

[20] P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry, Springer, New York, 2005 | MR | Zbl

[21] H. Maehara, “Embedding a polytope in a lattice”, Discrete Comput. Geom., 13:3-4 (1995), 585–592 | DOI | MR | Zbl

[22] A. M. Raigorodskii, “O distantsionnykh grafakh, imeyuschikh bolshoe khromaticheskoe chislo, no ne soderzhaschikh bolshikh simpleksov”, UMN, 62:6 (378) (2007), 187–188 | DOI | MR | Zbl

[23] A. M. Raigorodskii, O. I. Rubanov, “Small clique and large chromatic number”, European Conference on Combinatorics, Graph Theory and Applications, Electron. Notes Discrete Math., 34, Elsevier Sci. B. V., Amsterdam, 2009, 441–445 | DOI | MR | Zbl

[24] A. M. Raigorodskii, O. I. Rubanov, “On the clique and the chromatic numbers of high-dimensional distance graphs”, Number Theory and Applications, Hindustan Book Agency, New Delhi, 2009, 149–155 | MR | Zbl

[25] A. M. Raigorodskii, O. I. Rubanov, “O grafakh rasstoyanii s bolshim khromaticheskim chislom i bez bolshikh klik”, Matem. zametki, 87:3 (2010), 417–428 | DOI | MR | Zbl

[26] A. B. Kupavskii, A. M. Raigorodskii, “O distantsionnykh grafakh s bolshim khromaticheskim i malym klikovym chislami”, Dokl. AN, 444:5 (2012), 483–487 | MR | Zbl

[27] A. B. Kupavskii, A. M. Raigorodskii, “O prepyatstviyakh k realizatsii distantsionnykh grafov s bolshim khromaticheskim chislom na sferakh malogo radiusa”, Matem. sb., 204:10 (2013), 47–90 | DOI | MR | Zbl

[28] E. E. Demekhin, A. M. Raigorodskii, O. I. Rubanov, “Distantsionnye grafy, imeyuschie bolshoe khromaticheskoe chislo i ne soderzhaschie klik ili tsiklov zadannogo razmera”, Matem. sb., 204:4 (2013), 49–78 | DOI | MR | Zbl

[29] A. B. Kupavskii, “Distance graphs with large chromatic number and arbitrary girth”, Mosc. J. Comb. Number Theory, 2:2 (2012), 52–62 | MR | Zbl

[30] N. Alon, A. Kupavskii, “Two notions of unit distance graphs”, J. Combin. Theory Ser. A, 125 (2014), 1–17 | DOI | MR | Zbl

[31] A. B. Kupavskii, “Yavnye i veroyatnostnye konstruktsii distantsionnykh grafov s malenkim klikovym i bolshim khromaticheskim chislami”, Izv. RAN. Ser. matem., 78:1 (2014), 65–98 | DOI | MR | Zbl

[32] P. O'Donnell, “Arbitrary girth, 4-chromatic unit distance graphs in the plane. I. Graph description”, Geombinatorics, 9:3 (2000), 145–152 | MR | Zbl

[33] P. O'Donnell, “Arbitrary girth, 4-chromatic unit distance graphs in the plane. II. Graph embedding”, Geombinatorics, 9:4 (2000), 180–193 | MR | Zbl

[34] O. I. Rubanov, “Khromaticheskie chisla trekhmernykh grafov rasstoyanii, ne soderzhaschikh tetraedrov”, Matem. zametki, 82:5 (2007), 797–800 | DOI | MR | Zbl