Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space

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.

• Publication year

  2026

• Venue

  Unknown venue

• Publication date

  2026-02-20

• Fields of study

  Mathematics

• Identifiers
  arXiv 2602.17943
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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