Geodesic
Refinement of Lower Bounds of the Chromatic Number of a Space with Forbidden One-Color Triangles
Matematičeskie zametki, Tome 105 (2019) no. 3, pp. 349-363.

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The present paper deals with estimates of the chromatic number of a space with forbidden one-color triangles. New lower bounds for the quantity under study, which are sharper than all bounds obtained so far, are presented.
Keywords: Nelson–Erdős–Hadwiger problem, chromatic number of a space with forbidden one-color triangles, linear-algebraic method. 
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A. V. Bobu; A. E. Kupriyanov. Refinement of Lower Bounds of the Chromatic Number of a Space with Forbidden One-Color Triangles. Matematičeskie zametki, Tome 105 (2019) no. 3, pp. 349-363. http://geodesic.mathdoc.fr/item/MZM_2019_105_3_a2/

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

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

[3] P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry, Springer, New York, 2006 | MR

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

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

[6] R. I. Prosanov, A. M. Raigorodskii, A. A. Sagdeev, “Uluchsheniya teoremy Frankla–Redlya i geometricheskie sledstviya”, Dokl. AN, 475:2 (2017), 137–139 | DOI | Zbl

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

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

[9] L. A. Székely, “Erdős on unit distances and the Szemerédi–Trotter theorems”, Paul Erdős and His Mathematics, Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, 2002, 649–666 | MR

[10] A. M. Raigorodskii, “Coloring Distance Graphs and Graphs of Diameters”, Thirty Essays on Geometric Graph Theory, Springer, New York, 2013, 429–460 | MR | Zbl

[11] A. M. Raigorodskii, “Combinatorial geometry and coding theory”, Fund. Inform., 145:3 (2016), 359–369 | DOI | MR | Zbl

[12] P. Frankl, A. Kupavskii, “Erd{ő}s–Ko–Rado theorem for $\{0,\pm1\}$-vectors”, J. Combin. Theory Ser. A, 155 (2018), 157–179 | DOI | MR | Zbl

[13] P. Frankl, A. Kupavskii, “Intersection theorems for $\{0,\pm1\}$-vectors and $s$-cross-intersecting families”, Mosc. J. Comb. Number Theory, 7:2 (2017), 91–109 | MR

[14] P. Frankl, A. Kupavskii, “A size-sensitive inequality for cross-intersecting families”, European J. Combin., 62 (2017), 263–271 | DOI | MR | Zbl

[15] A. Ya. Kanel-Belov, V. A. Voronov, D. D. Cherkashin, “O khromaticheskom chisle ploskosti”, Algebra i analiz, 29:5 (2017), 68–89 | MR

[16] D. D. Cherkashin, A. M. Raigorodskii, “O khromaticheskikh chislakh prostranstv maloi razmernosti”, Dokl. AN, 472:1 (2017), 11–12 | MR | Zbl

[17] L. E. Shabanov, A. M. Raigorodskii, “Turanovskie otsenki dlya distantsionnykh grafov”, Dokl. AN, 475:3 (2017), 254–257 | MR

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

[19] Z. Füredi, J.-H. Kang, “Distance graphs on $\mathbb Z^n$ with $l_1$-norm”, Theoret. Comput. Sci., 319:1-3 (2004), 357–366 | MR | Zbl

[20] A. Kupavskiy, “On the chromatic number of $\mathbb R^n$ with an arbitrary norm”, Discrete Math., 311:6 (2011), 437–440 | DOI | MR | Zbl

[21] E. I. Ponomarenko, A. M. Raigorodskii, “Novaya nizhnyaya otsenka khromaticheskogo chisla ratsionalnogo prostranstva”, UMN, 68:5 (413) (2013), 183–184 | DOI | MR | Zbl

[22] 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

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

[24] L. Lovász, “Self-dual polytopes and the chromatic number of distance graphs on the sphere”, Acta Sci. Math. (Szeged), 45:1-4 (1983), 317–323 | MR | Zbl

[25] A. M. Raigorodskii, “On the chromatic numbers of spheres in $\mathbb R^n$”, Combinatorica, 32:1 (2012), 111–123 | DOI | MR | Zbl

[26] A. B. Kupavskii, “O raskraskakh sfer, vlozhennykh v $\mathbb R^n$”, Matem. sb., 202:6 (2011), 83–110 | DOI | MR | Zbl

[27] O. A. Kostina, A. M. Raigorodskii, “O nizhnikh otsenkakh khromaticheskogo chisla sfery”, Dokl. AN, 463:6 (2015), 639 | DOI | Zbl

[28] O. A. Kostina, A. M. Raigorodskii, “O novykh nizhnikh otsenkakh khromaticheskogo chisla sfery”, Tr. MFTI, 7:2 (2015), 20–26

[29] A. V. Berdnikov, A. M. Raigorodskii, “O khromaticheskom chisle evklidova prostranstva s dvumya zapreschennymi rasstoyaniyami”, Matem. zametki, 96:5 (2014), 790–793 | DOI | MR | Zbl

[30] A. V. Berdnikov, “Otsenka khromaticheskogo chisla evklidova prostranstva s neskolkimi zapreschennymi rasstoyaniyami”, Matem. zametki, 99:5 (2016), 783–787 | DOI | MR | Zbl

[31] A. V. Berdnikov, “Chromatic number with several forbidden distances in the space with the $l_q$-metric”, J. Math. Sci. (N.Y.), 227:4 (2017), 395–401 | DOI | MR | Zbl

[32] D. V. Samirov, A. M. Raigorodskii, “Kaskadnyi poisk proobrazov i sovpadenii: globalnaya i lokalnaya versii”, Matem. zametki, 93:1 (2013), 127–143 | DOI | MR | Zbl

[33] D. V. Samirov, A. M. Raigorodskii, “Novye nizhnie otsenki khromaticheskogo chisla prostranstva s zapreschennymi ravnobedrennymi treugolnikami”, Dinamicheskie sistemy, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 125, VINITI RAN, M., 2013, 252–268

[34] D. V. Samirov, A. M. Raigorodskii, “Novye otsenki khromaticheskogo chisla prostranstva s zapreschennymi ravnobedrennymi treugolnikami”, Dokl. AN, 456:3 (2014), 280–283 | DOI | Zbl

[35] D. V. Samirov, A. M. Raigorodskii, “Ob odnoi zadache, svyazannoi s optimalnoi raskraskoi prostranstva bez odnotsvetnykh ravnobedrennykh treugolnikov”, Tr. MFTI, 7:2 (2015), 39–50

[36] A. V. Bobu, A. E. Kupriyanov, A. M. Raigorodskii, “O chisle reber odnorodnogo gipergrafa s diapazonom razreshennykh peresechenii”, Probl. peredachi inform., 53:4 (2017), 16–42

[37] A. M. Raigorodskii, A. A. Kharlamova, “O sovokupnostyakh $(-1,0,1)$-vektorov s zapretami na velichiny poparnykh skalyarnykh proizvedenii”, Trudy po vektornomu i tenzornomu analizu, 29, Izd-vo Mosk. un-ta, M., 2013, 130–146