Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space
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2026
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| author | Istrate, Gabriel Zara, Catalin |
| author_facet | Istrate, Gabriel Zara, Catalin |
| contents | 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 $Ω\subseteq \mathbb{R}^n$ (hence constant, for connected $Ω$).
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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_17943 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space Istrate, Gabriel Zara, Catalin Combinatorics 05D10, 52C10, 54E52 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 $Ω\subseteq \mathbb{R}^n$ (hence constant, for connected $Ω$). 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. |
| title | Hypersphere-Based Restricting Conditions for Colorings of the Euclidean Space |
| topic | Combinatorics 05D10, 52C10, 54E52 |
| url | https://arxiv.org/abs/2602.17943 |