Homotopies and transcendental extensions in colouring problems

Abstract

The goal of this paper is to show some new applications and possible generalizations of the oriented volume method applied to the polytope colouring problems. This approach originated from the work of McLennan and Tourky ([1]), several sources attribute some role in the development of this technique to John Milnor as well. The advantage of this method lies in its geometric nature, appealing to the intuitions from the theory of manifolds and from elementary notions of the homotopy theory. We use some well-known algebraic formulas for quantities related to the volume of simplices (generalized shoelace formula, Cayley-Menger determinant).We prove new results using this method: 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.We also develop the technique of geometric realizations with algebraically independent (over the filed of real algebraic numbers) coordinates of vertices. This enables us to link the method of oriented volume and the proof of Ky Fan's lemma, and also between statements on linear combinations of multivariate polynomials and theorems about colourings of polytopes.We conclude with some remarks about possible further developments. Motivation and basic notions.In this paper we are concerened primarily with triangulated polytopes and colourings of their vertices. By a n-simplex ∆ n we understand a convex hull of n + 1 affinely independent points p 1 , . . . , p n+1 in a Euclidean space. By its d-face we mean a convex hull of d + 1 distinct points from {p 1 , . . . , p n+1 }. If we don't want to specify the exact number of these points, we simply speak of a face of a simplex.