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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 3, Pages 81–108 (Mi izv641)  

This article is cited in 18 scientific papers (total in 18 papers)

The problems of Borsuk and Grünbaum on lattice polytopes

A. M. Raigorodskii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study two classical problems of combinatorial geometry, the Borsuk problem on partitioning sets into parts of smaller diameter and the Grünbaum problem on covering sets by balls. We obtain new non-trivial upper bounds for the minimum number of parts of smaller diameter into which an arbitrary lattice polytope can be partitioned, as well as for the minimum number of balls of the same diameter by which any such polytope can be covered.


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English version:
Izvestiya: Mathematics, 2005, 69:3, 513–537

Bibliographic databases:

UDC: 514.17+519.174
MSC: 52B20, 05C15, 05D15
Received: 01.10.2003

Citation: A. M. Raigorodskii, “The problems of Borsuk and Grünbaum on lattice polytopes”, Izv. RAN. Ser. Mat., 69:3 (2005), 81–108; Izv. Math., 69:3 (2005), 513–537

Citation in format AMSBIB
\by A.~M.~Raigorodskii
\paper The problems of Borsuk and Gr\"unbaum on lattice polytopes
\jour Izv. RAN. Ser. Mat.
\yr 2005
\vol 69
\issue 3
\pages 81--108
\jour Izv. Math.
\yr 2005
\vol 69
\issue 3
\pages 513--537

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    This publication is cited in the following articles:
    1. A. M. Raigorodskii, “On the structure of distance graphs with large chromatic numbers”, Math. Notes, 80:3 (2006), 451–453  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. A. M. Raigorodskii, “Around Borsuk's Hypothesis”, Journal of Mathematical Sciences, 154:4 (2008), 604–623  mathnet  crossref  mathscinet  zmath  elib
    3. E. E. Demekhin, A. M. Raigorodskii, O. I. Rubanov, “Distance graphs having large chromatic numbers and containing no cliques or cycles of a given size”, Sb. Math., 204:4 (2013), 508–538  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. B. Kupavskii, A. M. Raigorodskii, “Obstructions to the realization of distance graphs with large chromatic numbers on spheres of small radii”, Sb. Math., 204:10 (2013), 1435–1479  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Mihály Hujter, Zsolt Lángi, “On the multiple Borsuk numbers of sets”, Isr. J. Math, 199:1 (2014), 219  crossref  mathscinet  zmath  scopus
    6. E. I. Ponomarenko, A. M. Raigorodskii, “New Upper Bounds for the Independence Numbers of Graphs with Vertices in $\{-1,0,1\}^n$ and Their Applications to Problems of the Chromatic Numbers of Distance Graphs”, Math. Notes, 96:1 (2014), 140–148  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. S. Gusev, “New Upper Bound for the Chromatic Numberof a Random Subgraph of a Distance Graph”, Math. Notes, 97:3 (2015), 326–332  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. V. V. Utkin, “Hamiltonian Paths in Distance Graphs”, Math. Notes, 97:6 (2015), 919–929  mathnet  crossref  crossref  mathscinet  isi  elib
    9. A. V. Bobu, O. A. Kostina, A. E. Kupriyanov, “Independence numbers and chromatic numbers of some distance graphs”, Problems Inform. Transmission, 51:2 (2015), 165–176  mathnet  crossref  isi  elib
    10. M. M. Pyaderkin, “On the stability of the Erdös-Ko-Rado theorem”, Dokl. Math, 91:3 (2015), 290  crossref  mathscinet  zmath  scopus
    11. L. I. Bogolubsky, A. S. Gusev, M. M. Pyaderkin, A. M. Raigorodskii, “Independence numbers and chromatic numbers of the random subgraphs of some distance graphs”, Sb. Math., 206:10 (2015), 1340–1374  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. A. V. Burkin, “Small subgraphs in random distance graphs”, Theory Probab. Appl., 60:3 (2016), 367–382  mathnet  crossref  crossref  mathscinet  isi  elib
    13. A. V. Burkin, “The threshold probability for the property of planarity of a random subgraph of a regular graph”, Russian Math. Surveys, 70:6 (2015), 1170–1172  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. S. N. Popova, “Zero-one law for random subgraphs of some distance graphs with vertices in $\mathbb Z^n$”, Sb. Math., 207:3 (2016), 458–478  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. A. V. Bobu, A. E. Kupriyanov, A. M. Raigorodskii, “Asymptotic study of the maximum number of edges in a uniform hypergraph with one forbidden intersection”, Sb. Math., 207:5 (2016), 652–677  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. S. N. Popova, “Zero-one laws for random graphs with vertices in a Boolean cube”, Siberian Adv. Math., 27:1 (2017), 26–75  mathnet  crossref  crossref  mathscinet  elib
    17. Raigorodskii A.M., “Combinatorial Geometry and Coding Theory*”, Fundam. Inform., 145:3 (2016), 359–369  crossref  mathscinet  zmath  isi  elib  scopus
    18. K. D. Kovalenko, A. M. Raigorodskii, “Systems of Representatives”, Math. Notes, 106:3 (2019), 372–377  mathnet  crossref  crossref  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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