Skip to main content
SpringerLink
Log in

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

A particular class of estimates related to the Nelson–Erdős–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ℓ1, certain series of distance graphs of small dimensions are considered. Independence numbers of such graphs are estimated by using the linear-algebraic method and combinatorial observations. This makes it possible to obtain certain lower bounds for the chromatic numbers of the spaces mentioned above and, for each case, specify a series of graphs leading to the strongest results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Soifer, Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of Its Creators (Springer, New York, 2009).

    MATH  Google Scholar 

  2. L. A. Székely, “Erdős on the unit distances and the Szemerédi–Trotter theorems,” in Paul Erdős and his Mathematics, II, Bolyai Soc. Math. Stud. (János Bolyai Math. Soc., Budapest, 2002), Vol. 11, pp. 649–666.

    Google Scholar 

  3. A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” in Discrete Geometry and Algebraic Combinatorics, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2014), Vol. 625, pp. 93–109.

    Google Scholar 

  4. A. M. Raigorodskii, “Coloring Distance Graphs and Graphs of Diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.

    Google Scholar 

  5. A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56 (1 (337)), 107–146 (2001) [RussianMath. Surveys 56 (1), 103–139 (2001)].

    MathSciNet  MATH  Google Scholar 

  6. A. M. Raigorodskii, “Combinatorial geometry and coding theory,” Fund. Inform. 145 (3), 359–369 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005).

    MATH  Google Scholar 

  8. K. Cantwell, “Finite Euclidean Ramsey theory,” J. Combin. Theory Ser. A 73 (2), 273–285 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  9. K. B. Chilakamarri, “The unit–distance graph problem: a brief survey and some new results,” Bull. Inst. Combin. Appl. 8, 39–60 (1993).

    MathSciNet  MATH  Google Scholar 

  10. J. Cibulka, “On the chromatic number of real and rational spaces,” Geombinatorics 18 (2), 53–65 (2008).

    MathSciNet  MATH  Google Scholar 

  11. G. Exoo and D. Ismailescu, On the Chromatic Number of Rn for Small Values of n, arXiv: http://arxiv. org/abs/1408. 2002 (2014).

    MATH  Google Scholar 

  12. G. Exoo, D. Ismailescu, and M. Lim, “On the chromatic number of R4,” Discrete Comput. Geom. 52 (2), 416–423 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Kahle and B. Taha, “New lower bounds on χ(Rd), d = 8,...,12,” Geombinatorics 24 (3), 109–116 (2015).

    MathSciNet  MATH  Google Scholar 

  14. A. B. Kupavskii and A. M. Raigorodskii, “On the chromatic number R9,” Fundam. Prikl. Mat. 14 (5), 139–154 (2008) [J. Math. Sci. (New York) 163 (6), 120–131 (2009)].

    Google Scholar 

  15. D. G. Larman and C. A. Rogers, “The realization distances within sets in Euclidean space,” Mathematika 19, 1–24 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  16. Z. Nagy, “A certain constructive estimate of the Ramsey number,” Mat. Lapok 23 (301–302), 26 (1972).

    MathSciNet  Google Scholar 

  17. O. Nechushtan, “Note on the space chromatic number,” DiscreteMath. 256 (1–2), 499–507 (2002).

    Google Scholar 

  18. D. D. Cherkashin and A. M. Raigorodskii, “On the chromatic numbers of the spaces of small dimension,” Dokl. Akad. Nauk 472 (1), 11–12 (2017).

    MathSciNet  Google Scholar 

  19. D. Cherkashin, A. Kulikov, and A. Raigorodskii, “On the chromatic numbers of small–dimensional Euclidean spaces,” Discrete Appl. Math. 243, 125–131 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. É. Guterman, V. K. Lyubimov, A. M. Raigorodskii, and A. S. Usachev, “On the independence number of distance graphs with vertices in −1, 0, 1n,” Mat. Zametki 86 (5), 794–796 (2009) [Math. Notes 86 (5), 744–746 (2009)].

    Article  MathSciNet  Google Scholar 

  21. V. K. Lyubimov and A. M. Raigorodskii, “Lower bounds for the independence numbers of some distance graphs with vertices in −1, 0, 1n,” Dokl. Akad. Nauk 427 (4), 458–460 (2009) [Dokl. Math. 80 (1), 547–549 (2009)].

    Google Scholar 

  22. A. M. Raigorodskii and A. B. Kupavskii, “On the chromatic numbers of small–dimensional Euclidean spaces,” in European Conference on Combinatorics, Graph Theory and Applications, Electron. Notes Discrete Math. (Elsevier Sci. B. V., Amsterdam, 2009), Vol. 34, pp. 435–439.

    Google Scholar 

  23. P. Frankl and R. Wilson, “Intersection theorems with geometric consequences,” Combinatorica 1 (4), 357–368 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. M. Raigorodskii, “On the chromatic number of a space,” Uspekhi Mat. Nauk 55 (2 (332)), 147–148 (2000) [Russian Math. Surveys 55 (2), 351–352 (2000)].

    Google Scholar 

  25. M. Benda and M. Perles, “Colorings of metric spaces,” Geombinatorics 9 (3), 113–126 (2000).

    MathSciNet  MATH  Google Scholar 

  26. Z. Füredi and J.–H. Kang, “Distance graphs on Zn with l1–norm,” Theoret. Comput. Sci. 319 (1–3), 357–366 (2004).

    Article  MathSciNet  Google Scholar 

  27. Y. Katznelson, “Chromatic numbers of Cayley graphs on Z and recurrence,” Combinatorica 21 (2), 211–219 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. B. Kupavskii, “On the coloring of spheres embedded in Rn,” Mat. Sb. 202 (6), 83–110 (2011) [Sb. Math. 202 (6), 859–886 (2011)].

    Article  MathSciNet  Google Scholar 

  29. A. B. Kupavskii, “On lifting a lower bound for the chromatic number Rn to higher dimensions,” Dokl. Akad. Nauk 429 (3), 305–308 (2009)

    Google Scholar 

  30. A. M. Raigorodskii, “On the chromatic number of a space with two forbidden distances,” Dokl. Akad. Nauk 408 (5), 601–604 (2006)

    MathSciNet  Google Scholar 

  31. N. G. Moshchevitin and A. M. Raigorodskii, “Colorings of the Space Rn with Several Forbidden Distances,” Mat. Zametki 81 (5), 733–743 (2007) [Math. Notes 81 (5), 656–664 (2007)].

    Article  MathSciNet  Google Scholar 

  32. A. M. Raigorodskii and I. M. Shitova, “Chromatic numbers of real and rational spaces with real or rational forbidden distances,” Mat. Sb. 199 (4), 107–142 (2008) [Sb. Math. 199 (4), 579–612 (2008)].

    Article  MathSciNet  MATH  Google Scholar 

  33. A. M. Raigorodskii and I. M. Shitova, “On the chromatic number of Euclidean space and the Borsuk problem,” Mat. Zametki 83 (4), 636–639 (2008) [Math. Notes 83 (4), 579–582 (2008)].

    Article  MathSciNet  MATH  Google Scholar 

  34. E. S. Gorskaya, I. M. Mitricheva, V. Yu. Protasov, and A. M. Raigorodskii, “Estimating the chromatic numbers of Euclidean space by convex minimization methods,” Mat. Sb. 200 (6), 3–22 (2009) [Sb. Math. 200 (6), 783–801 (2009)].

    Article  MATH  Google Scholar 

  35. A. M. Raigorodskii, “On the chromatic numbers of the spheres in Euclidean space,” Dokl. Akad. Nauk 432 (2), 174–177 (2010) [Dokl. Math. 81 (3), 379–382 (2010)].

    MathSciNet  Google Scholar 

  36. A. M. Raigorodskii, “On the chromatic numbers of spheres in Rn,” Combinatorica 32 (1), 111–123 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  37. O. A. Kostina and A. M. Raigorodskii, “On the lower bounds for the chromatic number of sphere,” Dokl. Akad. Nauk 463 (6), 639–641 (2015).

    MathSciNet  MATH  Google Scholar 

  38. A. V. Berdnikov and A. M. Raigorodskii, “On the chromatic number of Euclidean space with two forbidden distances,” Mat. Zametki 96 (5), 790–793 (2014) [Math. Notes 96 (5), 827–830 (2014)].

    Article  MathSciNet  MATH  Google Scholar 

  39. A. B. Kupavskii, “On the chromatic number of Rn with an arbitrary norm,” DiscreteMath. 311 (6), 437–440 (2011).

    Google Scholar 

  40. A. B. Kupavskii, “On the chromatic number of Rn with a set of forbidden distances,” Dokl. Akad. Nauk 435 (6), 740–743 (2010) [Dokl. Math. 82 (3), 963–966 (2010)].

    Google Scholar 

  41. E. I. Ponomarenko and A. M. Raigorodskii, “New lower bound for the chromatic number of a rational space with one and two forbidden distances,” Mat. Zametki 97 (2), 255–261 (2015) [Math. Notes 97 (2), 249–254 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Ya. Kanel’–Belov, V. A. Voronov, and D. D. Cherkashin, “On the chromatic number of infinitesimal plane layer,” Algebra Anal. 29 (5), 68–89 (2017) [St. PetersburgMath. J. 29 (5), 761–775 (2018)].

    MATH  Google Scholar 

  43. A. A. Sagdeev, “On the Frankl–Rödl theorem,” Izv. Ross. Akad. Nauk Ser. Mat. 82 (6), 128–157 (2018) [Izv. Math. 82 (6), 1196–1224 (2018)].

    MathSciNet  MATH  Google Scholar 

  44. A. A. Sagdeev, “Improved Frankl–Rödl theorem and some of its geometric consequences,” Probl. Peredachi Inf. 54 (2), 45–72 (2018) [Probl. Inf. Transmission 54 (2), 139–164 (2018)].

    MathSciNet  Google Scholar 

  45. A. A. Sagdeev, “Exponentially Ramsey sets,” Probl. Peredachi Inf. [Probl. Inf. Transmission] (in press).

  46. R. I. Prosanov, A. M. Raigorodskii, and A. A. Sagdeev, “Improvements of the Frankl–Rödl theorem and the geometric consequences,” Dokl. Akad. Nauk 475 (2), 137–139 (2017) [Dokl. Math. 96 (1), 336–338 (2017)].

    MATH  Google Scholar 

  47. R. I. Prosanov, “Upper bounds for the chromatic numbers of Euclidean spaces with forbidden Ramsey sets,” Mat. Zametki 103 (2), 248–257 (2018) [Math. Notes 103 (2), 243–250 (2018)].

    Article  MathSciNet  MATH  Google Scholar 

  48. A. M. Raigorodskii and A. A. Sagdeev, “On a bound in extremal combinatorics,” Dokl. Akad. Nauk 478 (3), 271–273 (2018) [Dokl. Math. 97 (1), 47–48 (2018)].

    MathSciNet  MATH  Google Scholar 

  49. A. V. Berdnikov, “Estimate for the chromatic number of Euclidean space with several forbidden distances,” Mat. Zametki 99 (5), 783–787 (2016) [Math. Notes 99 (5), 774–778 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  50. A. V. Berdnikov, “Chromatic number with several forbidden distances in the space with the q–metric,” J. Math. Sci. (N. Y. ) 227 (4), 395–401 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  51. A. M. Raigorodskii, “The chromatic number of a space with the metric lq,” UspekhiMat. Nauk 59 (5 (359)), 161–162 (2004) [RussianMath. Surveys 59 (5), 973–975 (2004)].

    MathSciNet  MATH  Google Scholar 

  52. L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, Part 1: Preliminary Version 2 (Department of Computer Science, The University of Chicago, 1992).

    Google Scholar 

  53. L. Bogolubsky, in arXiv: https://doi.org/https://github.com/LevBogolubsky/chrom_lower_linear_algebra.

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    L. I. Bogolyubsky & A. M. Raigorodskii

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, 141701, Russia

    A. M. Raigorodskii

  3. Adygeya State University, Maikop, 385016, Russia

    A. M. Raigorodskii

  4. Buryat State University, Ulan-Ude, 670000, Russia

    A. M. Raigorodskii

Authors
  1. L. I. Bogolyubsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Raigorodskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to L. I. Bogolyubsky or A. M. Raigorodskii.

Additional information

Russian Text © L. I. Bogolyubsky, A. M. Raigorodskii, 2019, published in Matematicheskie Zametki, 2019, Vol. 105, No. 2, pp. 187–213.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bogolyubsky, L.I., Raigorodskii, A.M. A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2. Math Notes 105, 180–203 (2019). https://doi.org/10.1134/S000143461901022X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S000143461901022X

Keywords

Search

Navigation