New upper bounds for the chromatic numbers of Euclidean spaces
Abstract
A coloring of Rn is \emph{proper for the forbidden distance segment} [1,ℓ] if no two points of the same color are at a distance from [1,ℓ]; the minimum number of colors is χ(Rn,[1,ℓ]), and ℓ=1 gives the classical chromatic number χ(Rn) of the Nelson--Hadwiger problem. We prove the new upper bounds χ(R4)≤43, χ(R5)≤132, χ(R7)≤1029, χ(R9)≤7203, χ(R10)≤45619, improving the previously known $49$, $140$, $1372$, $17253$ and 310; in particular, this refutes the conjecture of Arman, Bondarenko, Prymak and Radchenko that $49$ and $140$ are optimal among all lattice colorings of R4 and R5. The first four bounds come from explicit rational lattices --- an Eisenstein lattice in R4, a lattice in general position in R5, and laminations of the Eisenstein colorings E6*/343 and E8/2401 in R7 and R9 --- and each is reduced, by one verification protocol, to a finite list of inequalities between explicitly written rational numbers checked in exact arithmetic. The fifth bound is analytic: we prove that for every Eisenstein lattice Λ the distance between same-colored cells of (3+ω)Λ equals √(7/3) λ1(Λ), which gives the exact widths of all known colorings with 7n/2 colors, and a product rule ∑i1/di2≤1 for the widths of orthogonal products; together they yield 45619=2401·19, the first bound in R10 below 3n, as well as χ(R25)≤4·712 and χ(R26)≤19·712. We also show that no sublattice of E8 of index below $2401$ defines a proper coloring. All code, exact certificates and data are open.
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BibTeX
@misc{ivanov2026upper,
title = {{New upper bounds for the chromatic numbers of Euclidean spaces}},
author = {Leonid Ivanov and Nadezhda Glushkova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20436},
eprint = {2609.20436},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20436},
}
RIS
TY - UNPB
TI - New upper bounds for the chromatic numbers of Euclidean spaces
AU - Leonid Ivanov
AU - Nadezhda Glushkova
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20436
UR - https://arxiv.org/abs/2609.20436
AB - A coloring of R^n is \emph{proper for the forbidden distance segment} [1,ℓ] if no two points of the same color are at a distance from [1,ℓ]; the minimum number of colors is χ(R^n,[1,ℓ]), and ℓ=1 gives the classical chromatic number χ(R^n) of the Nelson--Hadwiger problem. We prove the new upper bounds χ(R^4)≤43, χ(R^5)≤132, χ(R^7)≤1029, χ(R^9)≤7203, χ(R^10)≤45619, improving the previously known $49$, $140$, $1372$, $17253$ and 3^10; in particular, this refutes the conjecture of Arman, Bondarenko, Prymak and Radchenko that $49$ and $140$ are optimal among all lattice colorings of R^4 and R^5. The first four bounds come from explicit rational lattices --- an Eisenstein lattice in R^4, a lattice in general position in R^5, and laminations of the Eisenstein colorings E_6^*/343 and E_8/2401 in R^7 and R^9 --- and each is reduced, by one verification protocol, to a finite list of inequalities between explicitly written rational numbers checked in exact arithmetic. The fifth bound is analytic: we prove that for every Eisenstein lattice Λ the distance between same-colored cells of (3+ω)Λ equals √(7/3) λ_1(Λ), which gives the exact widths of all known colorings with 7^n/2 colors, and a product rule ∑_i1/d_i^2≤1 for the widths of orthogonal products; together they yield 45619=2401·19, the first bound in R^10 below 3^n, as well as χ(R^25)≤4·7^12 and χ(R^26)≤19·7^12. We also show that no sublattice of E_8 of index below $2401$ defines a proper coloring. All code, exact certificates and data are open.
ER -