Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space

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.