Discrete & Computational Geometry

, Volume 49, Issue 3, pp 444–453 | Cite as

An Analogue of Gromov’s Waist Theorem for Coloring the Cube

Article

Abstract

It is proved that if we partition a d-dimensional cube into \(n^d\) small cubes and color the small cubes in \(m+1\) colors then there exists a monochromatic connected component consisting of at least \(f(d, m) n^{d-m}\) small cubes. Another proof of this result is given in Matdinov’s preprint (Size of components of a cube coloring, arXiv:1111.3911, 2011)

Keywords

Graph coloring Covering dimension Waist inequality 

References

  1. 1.
    Belov-Kanel, A., Ivanov-Pogodaev, I., Malistov, A., Kharitonov, M.: Colorings and clusters. In: 22nd Summer Conference International Mathematical Tournament of Towns. Teberda, Karachai-Cherkess, 2–10 Aug 2010, http://olympiads.mccme.ru/lktg/2010/2/2-1en.pdf (2010)
  2. 2.
    Gale, D.: The game of hex and the Brouwer fixed-point theorem. Am. Math. Mon. 86, 818–827 (1979)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gromov, M.: Singularities, expanders, and topology of maps. Part 2: from combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal. 20(2), 416–526 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Guth, L.: The waist inequality in Gromov’s work. http://math.mit.edu/~lguth/Exposition/waist.pdf.
  6. 6.
    Karasev, R.N.: A topological central point theorem. Topol. Appl. 159(3), 864–868 (2012)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Karasev, R.N.: A simpler proof of the Boros–Füredi–Bárány–Pach–Gromov theorem. Discrete Comput. Geom. 47(3), 492–495 (2012)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Matdinov, M.: Size of components of a cube coloring (preprint). http://arxiv.org/abs/1111.3911 (2011)
  9. 9.
    Matoušek, J., Přívětivý, A.: Large monochromatic components in two-colored grids. SIAM J. Discret. Math. 22(1), 295–311 (2008)MATHCrossRefGoogle Scholar
  10. 10.
    Memarian, Y.: On Gromov’s waist of the sphere theorem. J. Topol. Anal. 3(1), 7–36 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Sitnikov, K.: Über die Rundheit der Kugel. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 213–215 (1958)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Institute for Information Transmission Problems RASMoscowRussia
  3. 3.Laboratory of Discrete and Computational GeometryYaroslavl’ State UniversityYaroslavl’Russia

Personalised recommendations