Colorful versions of the Lebesgue, KKM, and Hex theorem

TL;DR

This paper generalizes classical topological theorems like Lebesgue, KKM, and Hex to colored convex polytopes using quasitoric manifolds, broadening their applicability in geometric and combinatorial contexts.

Contribution

It extends fundamental covering theorems to new classes of convex polytopes with colorings, utilizing quasitoric manifolds for a versatile approach.

Findings

Generalized Lebesgue theorem to colored convex polytopes

Extended KKM theorem to colored polytopes beyond simplices

Generalized Hex theorem to n-colorable simple polytopes

Abstract

Following and developing ideas of R. Karasev (Covering dimension using toric varieties, arXiv:1307.3437), 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 quasitoric manifolds offers great flexibility and versatility in applying the general method.