Multilabeled and topological versions of the Hex theorem

Abstract

The purpose of our paper is to give a purely combinatorial and algorithmic proof of the Hex theorem with multiple labels. Moreover we extend this result for a new class of complexes, called n-essential. We also present a topological version of the Hex theorem and explain its relation to the fixed point property, the Poincaré-Miranda theorem and the covering (Lebesgue) dimension.