{
  "abstract": "Given integers $c\\geq 2$ and $s\\geq 0$, let $\\mathsf{M}_3(c,s)$ denote the least integer such that every set of at least $\\mathsf{M}_3(c,s)$ points in the plane, no three on a line, colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. Further, let $λ_3(c)$ be the least integer such that $\\mathsf{M}_3(c,λ_3(c))<\\infty$. \\citet{colorempty} proved that, for every $c\\geq 2$, $$\\left\\lfloor\\frac{c-1}{2}\\right\\rfloor \\leq λ_3(c)\\leq c-2.$$ Later, \\citet{cravioto2019almost} improved the upper bound to $c-3$, for $c\\geq 4$. In this paper, we refine their argument to obtain the following asymptotic improvement: $$λ_3(c) \\leq c-\\sqrt{c\\log c}+o (\\sqrt{c\\log c} ),$$ for all sufficiently large $c$. We also show that every $c$-coloring of a sufficiently large Horton set contains a monochromatic triangle with at most $\\lfloor \\frac{c-1}{2} \\rfloor$ interior points. This shows that the aforementioned lower bound on $λ_3(c)$ is sharp within the class of Horton sets. We conclude with a conjecture on the large-color asymptotics of $λ_3(c)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12325",
  "author_total": 5,
  "authors": [
    {
      "name": "Bhaswar B. Bhattacharya"
    },
    {
      "name": "Sandip Das"
    },
    {
      "name": "Sk Samim Islam"
    },
    {
      "name": "Aashirwad Mohapatra"
    },
    {
      "name": "Saumya Sen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ct5bxmw89wqw5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12325",
  "fulltext_url": null,
  "id": "ct5bxmw89wqw5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12325",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213001203",
  "pdf_url": "https://arxiv.org/pdf/2609.12325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ct5bxmw89wqw5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.12325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12325"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Almost Empty Monochromatic Triangles With Many Colors",
  "title_text": "Almost Empty Monochromatic Triangles With Many Colors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/ct5bxmw89wqw5",
  "venue": "arXiv",
  "versions": []
}