Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homotopies and transcendental extensions in colouring problems

Wojciech Duliński

TL;DR

The paper develops a unified framework that blends the oriented volume method with geometric realizations using coordinates algebraically independent over the real algebraic numbers to study coloring problems on polytopes. By tracking the volume sum through a carefully constructed homotopy and invoking transcendence-based independence of simplex volumes, it derives parity and counting results for Sperner-type labelings on product spaces such as $\Delta^n \times \Delta^m$. It establishes new results including the non-draw property for the generalized Y game, a product-of-simplices triangulation theorem, and a multilabeled Ky Fan's lemma, while also providing new proofs of multilabeled Sperner's lemma and a generalized Atanassov conjecture. The approach leverages field-theoretic tools (transcendence bases) to ensure linear independence of volumes and determinants, enabling robust fixed-point-type conclusions in high dimensions and across product polytopes.

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.

Homotopies and transcendental extensions in colouring problems

TL;DR

The paper develops a unified framework that blends the oriented volume method with geometric realizations using coordinates algebraically independent over the real algebraic numbers to study coloring problems on polytopes. By tracking the volume sum through a carefully constructed homotopy and invoking transcendence-based independence of simplex volumes, it derives parity and counting results for Sperner-type labelings on product spaces such as . It establishes new results including the non-draw property for the generalized Y game, a product-of-simplices triangulation theorem, and a multilabeled Ky Fan's lemma, while also providing new proofs of multilabeled Sperner's lemma and a generalized Atanassov conjecture. The approach leverages field-theoretic tools (transcendence bases) to ensure linear independence of volumes and determinants, enabling robust fixed-point-type conclusions in high dimensions and across product polytopes.

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.

Paper Structure

This paper contains 9 sections, 16 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Description of the General Proof Technique
  4. Sperner's Lemma, Generalized Y Game and the Generalized Shoelace Formula
  5. Algebraically Independent Coordinates
  6. Atanassov Conjecture
  7. Ky Fan's Lemma
  8. Product of Two Simplices and Multilabeled Sperner's Lemma
  9. Declarations

Key Result

Theorem 1.1

\newlabelgra-y0 Let $\Delta^n$ be an $n$-dimensional simplex with a given triangulation. Let $V$ be the set of vertices of this triangulation. If $V$ is colored with $n$ colors, there exists a connected monochromatic subgraph that intersects every codimension 1 face of $\Delta^n$.

Figures (3)

  • Figure 1: In the picture above, the graph $G$ is depicted in red, the face $F_G$ in blue, and the vertex $O_G$ is indicated by the biggest red dot.
  • Figure 1: A comparison between two geometric realizations
  • Figure 1: A coloring of vertices assigned to the triangulation of a cube that gives exactly one simplex of the type $(2,2)$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Multilabeled Ky Fan's Lemma
  • Lemma 3.1
  • Proof 1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proof 2: Proof of Theorem \ref{['gra-y']}
  • Theorem 4.1
  • ...and 21 more