Tytuł artykułu
Autorzy
Wybrane pełne teksty z tego czasopisma
Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
In this paper, we overview three closely related problems: Nelson-Hadwiger problem on coloring spaces with forbidden monochromatics distances; Borsuk's problem on partitioning sets in spaces into parts of smaller diameter; problem of finding codes with forbidden Hamming distances.
Wydawca
Czasopismo
Rocznik
Tom
Strony
359--369
Opis fizyczny
Bibliogr. 81 poz., tab.
Twórcy
autor
- Moscow Institute of Physics and Technology, State University Dolgoprudny, Moscow, Russia
- Lomonosov Moscow State University, Moscow, Russia
Bibliografia
- [1] Soifer A. The Mathematical Coloring Book. Springer; 2009. ISBN-13: 978-0387746401, 10: 0387746404.
- [2] Hadwiger H. Ein Überdeckungssatz für den Euklidischen Raum. Portugaliae Math. 1944; 4:140–144.
- [3] Agarwal PK, Pach J. Combinatorial geometry. John Wiley and Sons Inc., New York; 1995.
- [4] Brass P, Moser W, Pach J. Research problems in discrete geometry. Springer; 2005. ISBN: 978-0-387-23815-9, 978-0-387-29929-7.
- [5] Klee V, Wagon S. Old and new unsolved problems in plane geometry and number theory. Math. Association of America; 1991. ISBN- 13:9780883853153, 10:0883853159.
- [6] Raigorodskii AM. Borsuk’s problem and the chromatic numbers of metric spaces. Russian Math. Surveys. 2001; 56(N1):103–139. doi: 10.1070/RM2001v056n01ABEH000358.
- [7] Raigorodskii AM. The chromatic numbers. Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, Russia; 2015. Second edition (book in Russian).
- [8] Raigorodskii AM. The linear algebra method in combinatorics. Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, Russia; 2015. Second edition (book in Russian).
- [9] Raigorodskii AM. Cliques and cycles in distance graphs and graphs of diameters. “Discrete Geometry and Algebraic Combinatorics”, AMS, Contemporary Mathematics. 2014; 625:93–109. doi:10.1090/conm/625/12493.
- [10] Raigorodskii AM. Coloring Distance Graphs and Graphs of Diameters. Thirty Essays on Geometric Graph Theory, J. Pach ed., Springer; 2013. p. 429–460. doi: 10.1007/978-1-4614-0110-0 23.
- [11] Székely LA. Erdös on unit distances and the Szemerédi – Trotter theorems. Paul Erdös and his Mathematics, Bolyai Series Budapest, J. Bolyai Math. Soc., Springer. 2002; 11:649–666.
- [12] Coulson D. A 15-coloring of 3-space omitting distance one. Discrete Math. 2002; 256:83–90. doi:10.1016/S0012-365X(01)00183-2.
- [13] Radoičić R, Tóth G. Note on the chromatic number of the space. Discrete and Computational Geometry: The Goodman – Pollack Festschrift, 695–698. (Algorithms and Combinatorics, 25), Springer; 2003. doi:10.1007/978-3-642-55566-4 32.
- [14] Nechushtan O. Note on the space chromatic number. Discrete Math. 2002; 256:499–507. doi: 10.1016/S0012-365X(00)00406-4.
- [15] Cantwell K. Finite Euclidean Ramsey theory. J. Combin. Theory A. 1996; 73(N2):273–285.
- [16] Ivanov LL. An estimate of the chromatic number of the space R4. Russian Math. Surveys. 2006; 61(N5):984–986. Available from: http://stacks.iop.org/0036-0279/61/i=5/a=L07.
- [17] Kupavskiy AB. On coloring spheres embedded into Rn. Sb. Math. 2011; 202(N6):83–110. doi: 10.1070/SM2011v202n06ABEH004169.
- [18] Cibulka J. On the chromatic number of real and rational spaces. Geombinatorics. 2008; 18(N2):53–66.
- [19] Larman DG, Rogers CA. The realization of distances within sets in Euclidean space. Mathematika. 1972; 19:1–24.
- [20] Kupavskiy AB, Raigorodskii AM. On the chromatic number of R9. J. of Math. Sci. 2009; 163(N6):720–731.
- [21] Kupavskiy AB. On lifting of estimation of chromatic number of Rn in higher dimension. Doklady Math. 2009; 429(N3):305–308.
- [22] Raigorodskii AM. On the chromatic number of a space. Russian Math. Surveys, 2000; 55(N2):351–352.
- [23] Exoo G, Ismailescu D, Lim M. On the Chromatic Number of R4. Discrete and Computational Geometry. 2014; 52(N2):416–423. doi: 10.1007/s00454-014-9612-7.
- [24] Exoo G, Ismailescu D. On the Chromatic Number of Rn for Small Values of n. arXiv:1408.2002.
- [25] Borsuk K. Drei Sätze über die n-dimensionale euklidische Sphäre. Fundamenta Math. 1933; 20:177–190.
- [26] Eggleston HG. Covering a three-dimensional set with sets of smaller diameter. J. London Math. Soc. 1955; 30:11–24.
- [27] Raigorodskii AM. Around Borsuk’s conjecture. Journal of Math. Sci. 2008; 154(N4):604–623. doi: 10.1007/s10958-008-9196-y.
- [28] Bondarenko A. On Borsuk’s conjecture for two-distance sets. Discrete and Comput. Geometry, 2014; 51:509–515. doi: 10.1007/s00454-014-9579-4.
- [29] Jenrich T, Brouwer A. A 64-dimensional two-distance counterexample to Borsuk’s conjecture. The Electronic Journal of Combinatorics. 2014; 21(N4).
- [30] Raigorodskii AM. On a bound in Borsuk’s problem. Russian Math. Surveys, 1999; 54(N2):453–454.
- [31] Bourgain J, Lindenstrauss J. On covering a set in Rd by balls of the same diameter. Geometric Aspects of Functional Analysis (J. Lindenstrauss and V. Milman, eds.), Lecture Notes in Math., 1469, Springer-Verlag, Berlin; 1991. p. 138–144. Available from: http://dx.doi.org/10.1007/BFb0089220#sthash.9xuQLj7h.dpuf.
- [32] Schramm O. Illuminating sets of constant width. Mathematika, 1988; 35:180–189.
- [33] Hadwiger H. Überdeckung einer Menge durch Mengen kleineren Durchmessers. Comm. Math. Helv. 1945/46; 18:73–75; Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers. Comm. Math. Helv. 1946/47; 19:72–73.
- [34] Knast R. An approximative theorem for Borsuk’s conjecture. Proc. Cambridge Phil. Soc. 1974; N1:75–76.
- [35] Rogers CA. Symmetrical sets of constant width and their partitions. Mathematika. 1971; 18:105–111.
- [36] Raigorodskii AM. Borsuk’s problem for (0,1)-polyhedra and cross-polytopes. Doklady Math. 2000; 61:256–259. Available from: http://stacks.iop.org/0036-0279/56/i=1/a=R03.
- [37] Raigorodskii AM. The Borsuk and Grünbaum problems for lattice polytopes. Izvestiya Math. 2005; 69(N3):513–537. Available from: http://stacks.iop.org/1064-5632/69/i=3/a=A03.
- [38] Boltyanski VG, Martini H, Soltan PS. Excursions into combinatorial geometry. Universitext, Springer, Berlin; 1997. ISBN- 978-3-540-61341-1, 978-3-642-59237-9.
- [39] Raigorodskii AM. Borsuk’s problem. Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, Russia; 2015. Second edition (book in Russian).
- [40] Raigorodskii AM. The Borsuk partition problem: the seventieth anniversary. Mathematical Intelligencer. 2004; 26(N3):4–12.
- [41] Bassalygo L, Cohen G, Zémor G. Codes with forbidden distances. Discrete Mathematics. 2000; 213:3–11.
- [42] Frankl P, Rödl V. Forbidden intersections. Trans. Amer. Math. Soc. 1987; 300(N1):259–286.
- [43] MacWilliams FJ, Sloane NJA. The theory of error-correcting codes. North-Holland, Amsterdam; 1977. ISBN- 13:978-0444851932, 10:0444851933.
- [44] Frankl P, Füredi Z. Forbidding just one intersection. Journal Combin. Theory A. 1985; 39:160–176.
- [45] Erdös P, Ko Ch, Rado R. Intersection theorems for systems of finite sets. J. Math. Oxford, Sec. 1961; 12(48).
- [46] Frankl P, Wilson R. Intersection theorems with geometric consequences . Combinatorica. 1981; 1:357–368.
- [47] Nagy Z. A certain constructive estimate of the Ramsey number (in Hungarian). Matematikai Lapok. 1972; 23, N 301-302, 26.
- [48] Bobu AV, Kostina OA, Kupriyanov AE. The independence numbers and the chromatic numbers of some distance graphs. Problems of Information Transmission. 2015; 51(N2):165–176. doi: 10.1134/S0032946015020076.
- [49] Raigorodskii AM. Systems of common representatives in combinatorics and their applications to geometry. Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, Russia; 2009. (book in Russian). ISBN- 10:5940575242, 13:978-5940575245.
- [50] Sidorenko AF. What we know and what we do not know about Tur´an numbers. Graphs and Combinatorics. 1995; 11(N2):179–199. doi: 10.1007/BF01929486.
- [51] Turán P. Egy gráfelmèleti szélsöértekfeladatrol. Mat. ´es Fiz. Lapok. 1941; 48(N3):436–453.
- [52] Matoušek J. Using the Borsuk – Ulam theorem. Universitext, Springer, Berlin; 2003. ISBN- 978-3-540-00362-5, 978-3-540-76649-0.
- [53] Balogh J, Kostochka AV, Raigorodskii AM. Coloring some finite sets in RN. Discussiones Mathematicae Graph Theory. 2013; 33(N1):25–31. doi: 10.7151/dmgt.1641.
- [54] Ponomarenko EI, Raigorodskii AM. An improvement to Frankl–Wilson’s theorem on the number of edges of a hypergraph with forbidden intersections. Doklady Math. 2014; 89(N1):59–60.
- [55] Ponomarenko EI, Raigorodskii AM. New estimates in the problem of the number of edges in a hypergraph with forbidden intersections. Problems of Information Transmission. 2013; 49(N4):384–390.
- [56] Bobu AV, Kupriyanov AE, Raigorodskii AM. On the maximum number of edges in a uniform hypergraph with one forbidden intersection. Doklady Math. 2015; 92(N1):401–403. doi: 10.1134/S106456241504002X.
- [57] Bobu AV, Kupriyanov AE, Raigorodskii AM. An asymptotic investigation of a problem concerning the maximum number of edges in a uniform hypergraph with one forbidden intersection. Sbornik Math. 2016; 207(N5).
- [58] Ahlswede R, Blinovsky VM. Lectures on advances in combinatorics. Springer; 2008. ISBN- 13:978-3540786016, 10:3540786015.
- [59] Ahlswede R, Khachatrian LH. The complete intersection theorem for systems of finite sets. Europ. J. Combin. 1997; 18:125–136.
- [60] Harlamova AA, Samirov DV, Raigorodskii AM, Zvonarev AE. An improvement to a theorem of Frankl and Rödl on the number of edges in a hypergraph with forbidden intersections. Doklady Math. 2014; 90(N1):432–434. doi: 10.1134/S1064562414050068.
- [61] Harlamova AA, Samirov DV, Raigorodskii AM, Zvonarev AE. On the chromatic number of a space with forbidden equilateral triangle. Sbornik Math. 2014; 205(N9):1310–1333.
- [62] Raigorodskii AM, Zvonarev AE. Improvements to Frankl–Rödl’s theorem on the number of edges in a hypergraph with forbidden intersections, and their applications to the problem of finding the chromatic number of a space with forbidden equilateral triangles. Proceedings of the Steklov Mathematical Institute. 2015; 288:94–104. doi: 10.1134/S0081543815010071.
- [63] Ziegler GM. Coloring Hamming graphs, optimal binary codes, and the 0/1 - Borsuk problem in low dimensions. Lect. Notes Comput. Sci. 2001; 2122:159–171. doi: 10.1007/3-540-45506-X 12.
- [64] Alon N, Spencer J. The probabilistic method.Wiley - Interscience Series in Discrete Math. and Optimization, Second Edition; 2000.
- [65] Bollobás B. Random Graphs. Cambridge Univ. Press, Second Edition; 2001.
- [66] Erdös P, Rényi A. On random graphs I. Publ. Math. Debrecen. 1959; 6:290–297.
- [67] Erdös P, Rényi A. On the evolution of random graphs. Bull. Inst. Int. Statist. Tokyo. 1961; 38:343–347.
- [68] Janson S, Łuczak T, Ruciński A. Random graphs. Wiley, NY; 2000.
- [69] Bogolyubskiy LI, Gusev AS, Pyaderkin MM, Raigorodskii AM. The independence numbers and the chromatic numbers of random subgraphs of some distance graphs. Doklady Math. 2014; 90(N1):462–465. doi:10.1134/S1064562414050147.
- [70] Bogolyubskiy LI, Gusev AS, Pyaderkin MM, Raigorodskii AM. The independence numbers and the chromatic numbers of random subgraphs of some distance graphs. Sbornik Math. 2015; 206(N10):3–36. Available from: http://stacks.iop.org/1064-5616/206/i=10/a=1340.
- [71] Bollobás B, Narayanan BP, Raigorodskii AM. On the stability of the Erdös–Ko–Rado theorem. J. Comb. Th. Ser. A. 2016; 137:64–78. doi: 10.1016/j.jcta.2015.08.002.
- [72] Kupavskii AB. On random subgraphs of Kneser and Schrijver graphs. J. Comb. Th., Ser. A. 2016; 141. arXiv:1502.00699.
- [73] Pyaderkin MM. Independence numbers of random subgraphs of distance graphs. Math. Notes. 2016; 99(N3):556–563. doi: 10.1134/S0001434616030299.
- [74] Pyaderkin MM. Independence numbers of random subgraphs of a distance graph. Math. Notes. 2016; 99(N2):288–297.
- [75] Gusev AS. New upper bound for the chromatic number of a random subgraph of a distance graph. Math. Notes. 2015; 97(N3):326–332. 10.1134/S0001434615030037.
- [76] Ahlswede R, Khachatrian LH. The complete nontrivial-intersection theorem for systems of finite sets. J. Combin. Theory A. 1996; 76:125–136. doi: 10.1006/eujc.1995.0092.
- [77] Erdös P, Rényi A. On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 1960; 5:17–61.
- [78] Raigorodskii AM. Three lectures on the Borsuk partition problem. London Mathematical Society Lecture Note Series. 2007; 347:202–248. Available from: http://dx.doi.org/10.1017/CBO9780511666315.007.
- [79] Raigorodskii AM. The Borsuk problem for integral polytopes. Sbornik Math. 2002; 193:1535–1556.
- [80] Raigorodskii AM. Borsuk’s problem for (0; 1)-polytopes and cross-polytopes. Doklady Math. 2002; 65:413–416.
- [81] Raigorodskii AM. The Borsuk, Grünbaum, and Hadwiger problems for some classes of polytopes and graphs. Doklady Math. 2003; 67(N1):85–89.
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-f87f7623-6500-47b9-9303-fe018ac074b9