Homotopies and transcendental extensions in colouring problems

Wojciech Duliński

Published 2020-11-24Version 1

We develop the technique of geometric realizations with algebraically independent (over the field of real algebraic numbers) coordinates of vertices and combine it with the oriented volume method inspired by work of McLennan and Tourky on the Sperner's lemma. This enables us to prove new results: the non-draw property of the generalized Y~game, the theorem about triangulation of the product of two simplices, the generalized Atanassov conjecture, and give new proofs of known results: the multilabeled versions of Sperner's and Ky Fan's lemmas.