{
  "abstract": "Let $K$ be a $1$-separated set of diameter at most $R-1$, and let $\\ell_m$ denote a collection of $m$ points on a line, with consecutive points of distance $1$ apart. Conlon and Fox (2019) demonstrated a coloring of $n$-dimensional Euclidean space avoiding red congruent copies of $\\ell_2$ and blue congruent copies of $K$ for $|K| > 10000^n\\log_2 R$. We show here a stronger bound, that in fact $|K| > (6.79 + o(1))^n\\log R$ suffices for arbitrary $1$-separated $K$, while the improvement $|K| > (5 + o(1))^n\\log R$ holds in many cases, including when $K = \\ell_m$, or more generally when $K$ is contained in a low-dimensional affine subspace. We also make a special study of the case when $n=2$, demonstrating a two-coloring of two-dimensional Euclidean space avoiding red copies of $\\ell_2$ and blue copies of $\\ell_{6330}$. This latter result addresses a question of Erdős and Graham.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.17194",
  "author_total": 4,
  "authors": [
    {
      "name": "Gabriel Currier"
    },
    {
      "name": "Param Mody"
    },
    {
      "name": "Zehan Xie"
    },
    {
      "name": "Jiaming Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dt5g5wx71ymzf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.17194",
  "fulltext_url": null,
  "id": "dt5g5wx71ymzf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.17194",
  "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": "W7165009991",
  "pdf_url": "https://arxiv.org/pdf/2606.17194",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dt5g5wx71ymzf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2606.17194",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.17194"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory",
  "title_text": "Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/dt5g5wx71ymzf",
  "venue": "arXiv",
  "versions": []
}