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Chessboard and level sets of continuous functions

Michał Dybowski, Przemysław Górka

TL;DR

The paper develops a discrete, grid-based framework linking $m$-distance clustered colorings on $\mathcal{K}_k^n$ to topological connectivity across opposite faces of the cube, establishing a bound $|P|\le C_{n,m}$ for a $1$-connected color class whose preimage yields a connected set connecting opposite faces. This combinatorial result is then lifted to the continuous setting: for every continuous $f:I^n\to\mathbb{R}^{n-1}$ there exists a scalar $p$ and a compact $S\subset f^{-1}({p})$ connecting opposite faces, yielding a direct route to the Poincaré-Miranda and Brouwer Fixed Point Theorems and to the $n$-dimensional Steinhaus Chessboard Theorem. The work also investigates minimal constants, provides explicit bounds such as $\widehat{C}_{n,m}\le (n-1)!(m+1)^{n-1}$, and shows that the result cannot be extended to path-connected sets in general. Hausdorff-convergence techniques are used to pass from the discrete approximation to the exact continuous conclusion, highlighting the tight interplay between combinatorial coloring and topological fixed-point results.

Abstract

We provide the following result and its discrete equivalent: Let $f \colon I^n \to \mathbb{R}^{n-1}$ be a continuous function. Then, there exist a point $p \in \mathbb{R}^{n-1}$ and a compact subset $S \subset f^{-1}\left[\left\{p\right\}\right]$ which connects some opposite faces of the $n$-dimensional unit cube $I^n$. We give an example that shows it cannot be generalized to path-connected sets. Additionally, we show that the $n$-dimensional Steinhaus Chessboard Theorem and the Brouwer Fixed Point Theorem are simple consequences of this result.

Chessboard and level sets of continuous functions

TL;DR

The paper develops a discrete, grid-based framework linking -distance clustered colorings on to topological connectivity across opposite faces of the cube, establishing a bound for a -connected color class whose preimage yields a connected set connecting opposite faces. This combinatorial result is then lifted to the continuous setting: for every continuous there exists a scalar and a compact connecting opposite faces, yielding a direct route to the Poincaré-Miranda and Brouwer Fixed Point Theorems and to the -dimensional Steinhaus Chessboard Theorem. The work also investigates minimal constants, provides explicit bounds such as , and shows that the result cannot be extended to path-connected sets in general. Hausdorff-convergence techniques are used to pass from the discrete approximation to the exact continuous conclusion, highlighting the tight interplay between combinatorial coloring and topological fixed-point results.

Abstract

We provide the following result and its discrete equivalent: Let be a continuous function. Then, there exist a point and a compact subset which connects some opposite faces of the -dimensional unit cube . We give an example that shows it cannot be generalized to path-connected sets. Additionally, we show that the -dimensional Steinhaus Chessboard Theorem and the Brouwer Fixed Point Theorem are simple consequences of this result.
Paper Structure (7 sections, 21 theorems, 22 equations, 4 figures)

This paper contains 7 sections, 21 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $m$-distance clustered chromatic number
  4. The main combinatorial result
  5. Minimum constants
  6. The continuous result
  7. Consequences of the continuous result

Key Result

Theorem A

Let $n \in \mathbb{N}, m \in \mathbb{Z}_+$ be such that $0 \le m \le n-1$. Then, there exists a constant $C_{n, m}>0$ such that the following property holds:

Figures (4)

  • Figure 1: Graph $G_2$.
  • Figure 2: A $2$-distance clustered $3$-coloring of $G_2$.
  • Figure 3: An illustration of a generic situation from the proof.
  • Figure 4: The graph of the function $g$.

Theorems & Definitions (41)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Theorem 2.2: $n$-dimensional Steinhaus Chessboard Theorem, chessboard
  • Definition 2.3
  • proof
  • Proposition 2.5
  • proof
  • Lemma 2.6: kuratowski
  • Theorem 2.7: Czarnecki
  • ...and 31 more