Abstract
In 1933, Borsuk stated a conjecture, which has become classical, that the minimum number of parts of smaller diameter into which an arbitrary set of diameter 1 in \(\mathbb{R}^n\) can be partitioned is \(n+1\). In 1993, this conjecture was disproved using sets of points with coordinates 0 and 1. Later, the second author obtained stronger counterexamples based on families of points with coordinates \(-1\), \(0\), and \(1\). We establish new lower bounds for Borsuk numbers in families of this type.
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References
Raigorodskii, A.M., Cliques and Cycles in Distance Graphs and Graphs of Diameters, in Discrete Geometry and Algebraic Combinatorics (AMS Special Session on Discrete Geometry and Algebraic Combinatorics, San Diego, CA, USA, Jan. 11, 2013), Providence, RI: Amer. Math. Soc., 2014, pp. 93–109.
Raigorodskii, A.M., Combinatorial Geometry and Coding Theory, Fund. Inform., 2016, vol. 145, no. 3, pp. 359–369. https://doi.org/10.3233/FI-2016-1365
Raigorodskii, A.M., Borsuk's Problem and the Chromatic Numbers of Some Metric Spaces, Uspekhi Mat. Nauk, 2001, vol. 56, no. 1,(337), pp. 107–146 [Russian Math. Surveys (Engl. Transl.), 2001, vol. 56, no. 1, pp. 103–139]. https://doi.org/10.1070/RM2001v056n01ABEH000358
Raigorodskii, A.M., Around Borsuk's Hypothesis, Sovrem. Mat. Fundam. Napravl., 2007, vol. 23, pp. 147–164 [J. Math. Sci. (N.Y.) (Engl. Transl.), 2007, vol. 154, no. 4, pp. 604–623]. https://doi.org/10.1007/s10958-008-9196-y
Bourgain, J. and Lindenstrauss, J., On Covering a Set in \(\mathbb{R}^d\) by Balls of the Same Diameter, Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 1989–90, Lindenstrauss, J. and Milman, V.D., Eds., Lect. Notes Math., vol. 1469, Berlin: Springer-Verlag, 1991, pp. 138–144. https://doi.org/10.1007/BFb0089220
Schramm, O., Illuminating Sets of Constant Width, Mathematika, 1988, vol. 35, no. 2, pp. 180–189. https://doi.org/10.1112/S0025579300015175
Bogolyubsky, L.I. and Raigorodskii, A.M., A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics \(\ell_1\) and \(\ell_2\), Mat. Zametki, 2019, vol. 105, no. 2, pp. 187–213 [Math. Notes (Engl. Transl.), 2019, vol. 105, no. 1–2, pp. 180–203]. https://doi.org/10.1134/S000143461901022X
Raigorodskii, A.M. and Bogolyubsky, L.I., On Bounds in Borsuk's Problem, Tr. Mosk. Fiz.-Tekh. Inst., 2019, vol. 11, no. 3, pp. 20–49.
Filimonov, V.P., Covering sets in \(\mathbb{R}^m\), Mat. Sb., 2014, vol. 205, no. 8, pp. 95–138 [Sb. Math. (Engl. Transl.), 2014, vol. 205, no. 8, pp. 1160–1200]. https://doi.org/10.1070/SM2014v205n08ABEH004414
Raigorodskii, A.M. and Koshelev, M.M., New Bounds for the Clique-Chromatic Numbers of Johnson Graphs, Dokl. Ross. Akad. Nauk, 2020, vol. 490, no. 1, pp. 78–80 [Dokl. Math. (Engl. Transl.), 2020, vol. 101, no. 1, pp. 66–67]. https://doi.org/10.1134/S1064562420010184
Ipatov, M.M., Koshelev, M.M., and Raigorodskii, A.M., Modularity of Some Distance Graphs, Dokl. Ross. Akad. Nauk, 2020, vol. 490, no. 1, pp. 71–73 [Dokl. Math. (Engl. Transl.), 2020, vol. 101, no. 1, pp. 60–61]. https://doi.org/10.1134/S1064562420010147
Raigorodskii, A.M. and Koshelev, M.M., New Bounds on Clique-Chromatic Numbers of Johnson Graphs, Discrete Appl. Math., 2020, vol. 283, pp. 724–729. https://doi.org/10.1016/j.dam.2020.01.015
Pushnyakov, F.A. and Raigorodskii, A.M., Estimate of the Number of Edges in Special Subgraphs of a Distance Graph, Mat. Zametki, 2020, vol. 107, no. 2, pp. 286–298 [Math. Notes (Engl. Transl.), 2020, vol. 107, no. 1–2, pp. 322–332]. https://doi.org/10.1134/S0001434620010320
Raigorodskii, A.M. and Kharlamova, A.A., On Sets of \((-1,0,1)\)-Vectors with Prohibited Values of Pairwise Inner Products, Tr. Sem. Vect. Tenz. Anal., vol. 29, Moscow: Moscow State Univ., 2013, pp. 130–146.
Frankl, P. and Kupavskii, A., Erdős–Ko–Rado Theorem for \(\{0,\pm1\}\)-Vectors, J. Combin. Theory Ser. A, 2018, vol. 155, pp. 157–179. https://doi.org/10.1016/j.jcta.2017.11.003
Frankl, P. and Kupavskii, A., Incompatible Intersection Properties, Combinatorica, 2019, vol. 39, no. 6, pp. 1255–1266. https://doi.org/10.1007/s00493-019-4064-6
Kupavskii, A., Degree Versions of Theorems on Intersecting Families via Stability, J. Combin. Theory Ser. A, 2019, vol. 168, pp. 272–287. https://doi.org/10.1016/j.jcta.2019.06.002
Ihringer, F. and Kupavskii, A., Regular Intersecting Families, Discrete Appl. Math., 2019, vol. 270, pp. 142–152. https://doi.org/10.1016/j.dam.2019.07.009
Bobu, A.V., Kupriyanov, A.É., and Raigorodskii, A.M., A Generalization of Kneser Graphs, Mat. Zametki, 2020, vol. 107, no. 3, pp. 351–365 [Math. Notes (Engl. Transl.), 2020, vol. 107, no. 3–4, pp. 392–403]. https://doi.org/10.1134/S0001434620030037
Sagdeev, A.A. and Raigorodskii, A.M., On a Frankl–Wilson Theorem and Its Geometric Corollaries, Acta Math. Univ. Comenian. (N.S.), 2019, vol. 88, no. 3, pp. 1029–1033.
Raigorodskii, A.M. and Shishunov, E.D., On the Independence Numbers of Some Distance Graphs with Vertices in \(\{-1,0,1\}^n\), Dokl. Akad. Nauk, 2019, vol. 485, no. 3, pp. 269–271 [Dokl. Math. (Engl. Transl.), 2019, vol. 99, no. 2, pp. 165–166]. https://doi.org/10.1134/S1064562419020194
Raigorodskii, A.M. and Shishunov, E.D., On the Independence Numbers of Distance Graphs with Vertices in \(\{-1,0,1\}^n\), Dokl. Akad. Nauk, 2019, vol. 488, no. 5, pp. 486–487 [Dokl. Math. (Engl. Transl.), 2019, vol. 100, no. 2, pp. 476–477]. https://doi.org/10.1134/S1064562419050193
Sokolov, A.A. and Raigorodskiy, A.M., On Rational Analogs of the Nelson–Hadwiger and Borsuk Problems, Chebyshevskii Sb., 2018, vol. 19, no. 3, pp. 270–281. https://doi.org/10.22405/2226-8383-2018-19-3-270-281
Raigorodskii, A.M. and Trukhan, T.V., On the Chromatic Numbers of Some Distance Graphs, Dokl. Akad. Nauk, 2018, vol. 482, no. 6, pp. 648–650 [Dokl. Math. (Engl. Transl.), 2018, vol. 98, no. 2, pp. 515–517]. https://doi.org/10.1134/S1064562418060297
Cherkashin, D., Kulikov, A., and Raigorodskii, A., On the Chromatic Numbers of Small-Dimensional Euclidean Spaces, Discrete Appl. Math., 2018, vol. 243, pp. 125–131. https://doi.org/10.1016/j.dam.2018.02.005
Raigorodskii, A.M. and Sagdeev, A.A., On a Bound in Extremal Combinatorics, Dokl. Akad. Nauk, 2018, vol. 478, no. 3, pp. 271–273 [Dokl. Math. (Engl. Transl.), 2018, vol. 97, no. 1, pp. 47–48]. https://doi.org/10.1134/S1064562418010155
Sagdeev, A.A., On a Frankl–Wilson Theorem, Probl. Peredachi Inf., 2019, vol. 55, no. 4, pp. 86–106 [Probl. Inf. Transm. (Engl. Transl.), 2019, vol. 55, no. 4, pp. 376–395]. https://doi.org/10.1134/S0032946019040045
Cherkashin, D. and Kiselev, S., Independence Numbers of Johnson-type Graphs, arXiv:1907.06752 [math.CO], 2019.
Zakharov, D.A., Chromatic Numbers of Some Distance Graphs, Mat. Zametki, 2020, vol. 107, no. 2, pp. 210–220 [Math. Notes (Engl. Transl.), 2020, vol. 107, no. 1–2, pp. 238–246]. https://doi.org/10.1134/S000143462001023X
Zakharov, D., Chromatic Numbers of Kneser-type Graphs, J. Combin. Theory Ser. A, 2020, vol. 172, pp. 105188 (16 pp.). https://doi.org/10.1016/j.jcta.2019.105188
Prosanov, R.I., Counterexamples to Borsuk's Conjecture with Large Girth, Mat. Zametki, 2019, vol. 105, no. 6, pp. 890–898 [Math. Notes (Engl. Transl.), 2019, vol. 105, no. 5–6, pp. 874–880]. https://doi.org/10.1134/S0001434619050249
Baker, R.C., Harman, G., and Pintz, J., The Difference between Consecutive Primes. II, Proc. London Math. Soc. (3), 2001, vol. 83, no. 3, pp. 532–562. https://doi.org/10.1112/plms/83.3.532
Funding
The research was supported in part by the Russian Foundation for Basic Research, project no. 18-01-00355, and the President of the Russian Federation Council for State Support of Leading Scientific Schools, grant no. NSh-2540.2020.1.
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Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 2, pp. 44–50 https://doi.org/10.31857/S0555292321020030.
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Berdnikov, A., Raigorodskii, A. Bounds on Borsuk Numbers in Distance Graphs of a Special Type. Probl Inf Transm 57, 136–142 (2021). https://doi.org/10.1134/S0032946021020034
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DOI: https://doi.org/10.1134/S0032946021020034