Colorful versions of the Lebesgue, KKM, and Hex theorem

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.