An Upper Bound of (3√3/2 + o(1))^d for the Chromatic Number of Euclidean Space
Abstract
Preprint. We prove that Euclidean space admits a periodic Borel coloring with no monochromatic unit-distance pair and at most C d log d (3√3/2)^d colors, for an absolute constant C and every integer d ≥ 2. In particular, limsup χ(R^d)^{1/d} ≤ 3√3/2. The classical Larman–Rogers bound is (3+o(1))^d. Together with the companion lower bound χ(R^d) ≥ c C*^d (C* > 1.309251), the exponential base lies in the interval [1.30925, 3√3/2]. The upload contains the article, the source, verification scripts, and a one-page extended abstract.
Fields of Study & Subjects
Limits and Structures in Graph Theory
Computational Geometry and Mesh Generation
Point processes and geometric inequalities
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 19 September 2026
- Language
- EN
- Document type
- Preprint