Discrete & Computational Geometry

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

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



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)


Graph coloring Covering dimension Waist inequality 


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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

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