Homotopies and transcendental extensions in colouring problems

TL;DR

This paper introduces a novel geometric approach combining algebraically independent coordinates and volume methods to prove new results and provide alternative proofs in combinatorial coloring problems.

Contribution

It develops a new technique that merges algebraic independence with volume methods, leading to new proofs and results in combinatorial topology and coloring problems.

Findings

Proves the non-draw property of the generalized Y game.

Establishes a triangulation theorem for the product of two simplices.

Provides new proofs for Sperner's lemma and Atanassov's conjecture.

Abstract

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, multilabeled Ky Fan' s lemma, and give new proofs of known results: the multilabeled version of Sperner's lemma and generalized Atanassov conjecture.