{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/upper-bounds-on-chromatic-number-of-mathbb-e","title":"Upper bounds on chromatic number of $\\mathbb{E}^n$ in low dimensions","arxiv_id":"2112.13438","date":"2021-12-26","proceeding":null,"authors":["Andrii Arman","Andriy V. Bondarenko","Andriy Prymak","Danylo Radchenko"],"abstract":"Let $\\chi(\\mathbb{E}^n)$ denote the chromatic number of the Euclidean space $\\mathbb{E}^n$, i.e., the smallest number of colors that can be used to color $\\mathbb{E}^n$ so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of $\\mathbb{E}^n$ based on sublattice coloring schemes that establish the following new bounds: $\\chi(\\mathbb{E}^5)\\le 140$, $\\chi(\\mathbb{E}^n)\\le 7^{n/2}$ for $n\\in\\{6,8,24\\}$, $\\chi(\\mathbb{E}^7)\\le 1372$, $\\chi(\\mathbb{E}^{9})\\leq 17253$, and $\\chi(\\mathbb{E}^n)\\le 3^n$ for all $n\\le 38$ and $n=48,49$.","url_abs":"https://arxiv.org/abs/2112.13438v2","url_pdf":"https://arxiv.org/pdf/2112.13438v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"upper-bounds-on-chromatic-number-of-mathbb-e","repo_url":"https://github.com/andriyprm/ubcnessd","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}