Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space

Gabriel Istrate, Catalin Zara

Abstract

We investigate colorings of the Euclidean space \(\mathbb{R}^n\) in which the color of a point \(p\) is determined by monochromatic configurations of points lying on hyperspheres centered at \(p\). We consider two types of conditions: those based on the cardinality of the forcing sets and admissible radii, and those based on certain geometric properties of simplices, such as shape, edge lengths, or volumes, for colorings using countably (finite or infinite) many colors. Our main objective is to determine whether a given condition forces the coloring to be monochromatic or not. Examples constructed using existing results show that, for conditions based on shapes, additional regularity assumptions on color classes are necessary. Accordingly, we study colorings that are somewhere comeager.

Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space

Abstract

We investigate colorings of the Euclidean space in which the color of a point is determined by monochromatic configurations of points lying on hyperspheres centered at . We consider two types of conditions: those based on the cardinality of the forcing sets and admissible radii, and those based on certain geometric properties of simplices, such as shape, edge lengths, or volumes, for colorings using countably (finite or infinite) many colors. Our main objective is to determine whether a given condition forces the coloring to be monochromatic or not. Examples constructed using existing results show that, for conditions based on shapes, additional regularity assumptions on color classes are necessary. Accordingly, we study colorings that are somewhere comeager.
Paper Structure (7 sections, 11 theorems, 30 equations)

This paper contains 7 sections, 11 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Cardinality Conditions
  3. Rigid Geometric Properties
  4. Somewhere Comeager Colorings
  5. Simplex Shape Conditions
  6. Edge-Length Conditions
  7. Volume Conditions

Key Result

Theorem 2.1

Let $\mathcal{R} \subset (0,\infty)$ be an uncountable set of admissible radii, let $\mathcal{C}$ be a set of colors of cardinality ${X} \leqslant \aleph_0$, let ${Y} \leqslant \aleph_0$ be a fixed cardinality, and let $\Omega \subset \mathbb{R}^n$ be a nonempty set of admissible centers. Let $f \co Let $f|_{\Omega} \colon \Omega \to \mathcal{C}$ denote the restriction of $f$ to the induced topolo

Theorems & Definitions (29)

  • Theorem 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Definition 4.1
  • Proposition 4.2
  • ...and 19 more