Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space

TL;DR

This paper investigates how hypersphere forcing conditions influence colorings of Euclidean space, demonstrating conditions under which such colorings become locally or globally monochromatic, especially under regularity assumptions and geometric constraints.

Contribution

It introduces new theorems linking hypersphere forcing conditions with local and global monochromaticity in Euclidean space colorings, including regularity and geometric property considerations.

Findings

Countably many colors and uncountable radii lead to local monochromaticity.

Rigid geometric properties require regularity assumptions for forcing conditions to imply monochromaticity.

Baire regularity and uniform cap conditions can ensure global monochromaticity.

Abstract

We study colorings of the Euclidean space constrained by \emph{hypersphere forcing conditions}: if an admissible hypersphere, $S_r(p)$, centered at a point $p$ and of radius $r$ contains a monochromatic set of points satisfying a certain property $\mathcal{P}$, then the center of the hypersphere must have that color. These forcing conditions may be restricted in applicability to a specific set of hyperspheres $S_r(p)$. For cardinality-based forcing conditions we prove a general theorem: for countably many colors and any uncountable set of admissible radii $\mathcal{R}$, such a coloring is locally monochromatic on any admissible center set $\Omega \subseteq \mathbb{R}^n$ (hence constant, for connected $\Omega$). For rigid geometric properties (simplex shape, edge-length, volume constraints) we show that forcing conditions alone are insufficient without regularity assumptions. Our main result shows that for colorings satisfying a certain Baire regularity condition rigid geometric properties enforce local monochromaticity and, in the presence of a certain \emph{``uniform cap" condition}, global monochromaticity. Applications include dichotomies for edge-length and volume constraints in terms of $\inf(\mathcal{L})$ and $\inf(\mathcal{V})$, and a comeagerness criterion in the ``all edges in $\mathcal{L}$'' regime.