{
  "abstract": "A coloring of $\\R^n$ is \\emph{proper for the forbidden distance segment} $[1,\\ell]$ if no two points of the same color are at a distance from $[1,\\ell]$; the minimum number of colors is $χ(\\R^n,[1,\\ell])$, and $\\ell=1$ gives the classical chromatic number $χ(\\R^n)$ of the Nelson--Hadwiger problem. We prove the new upper bounds $χ(\\R^4)\\le43$, $χ(\\R^5)\\le132$, $χ(\\R^7)\\le1029$, $χ(\\R^9)\\le7203$, $χ(\\R^{10})\\le45619$, 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 $\\sqrt{7/3}\\,λ_1(Λ)$, which gives the exact widths of all known colorings with $7^{n/2}$ colors, and a product rule $\\sum_i1/d_i^2\\le1$ for the widths of orthogonal products; together they yield $45619=2401\\cdot19$, the first bound in $\\R^{10}$ below $3^n$, as well as $χ(\\R^{25})\\le4\\cdot7^{12}$ and $χ(\\R^{26})\\le19\\cdot7^{12}$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20436",
  "author_total": 2,
  "authors": [
    {
      "name": "Leonid Ivanov"
    },
    {
      "name": "Nadezhda Glushkova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/225ksmq21dg00.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20436",
  "fulltext_url": null,
  "id": "225ksmq21dg00",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20436",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213604108",
  "pdf_url": "https://arxiv.org/pdf/2609.20436",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/225ksmq21dg00.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20436",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20436"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "New upper bounds for the chromatic numbers of Euclidean spaces",
  "title_text": "New upper bounds for the chromatic numbers of Euclidean spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/225ksmq21dg00",
  "venue": "arXiv",
  "versions": []
}