Authors: Baralić, Đorđe 
Živaljević, Rade 
Title: Colorful versions of the Lebesgue, KKM, and Hex theorem
Journal: Journal of Combinatorial Theory. Series A
Volume: 146
First page: 295
Last page: 311
Issue Date: 1-Feb-2017
Rank: M21
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2016.10.002
Abstract: 
Following and developing ideas of R. Karasev (2014) [10], we extend the Lebesgue theorem (on covers of cubes) and the Knaster–Kuratowski–Mazurkiewicz theorem (on covers of simplices) to different classes of convex polytopes (colored in the sense of M. Joswig). We also show that the n-dimensional Hex theorem admits a generalization where the n-dimensional cube is replaced by a n-colorable simple polytope. The use of specially designed quasitoric manifolds, with easily computable cohomology rings and the cohomological cup-length, offers a great flexibility and versatility in applying the general method.
Keywords: Cup-length | KKM theorem | Lebesgue theorem | Quasitoric manifolds | Simple polytopes
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 

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