{
  "abstract": "For integers $r \\ge 3$, $c \\ge 2$, and $s \\ge 0$, let $\\mathsf{M}_r^{\\star}(c,s)$ denote the smallest integer such that any set of at least $\\mathsf{M}_r^{\\star}(c,s)$ points in the plane in general position (no three on a line), colored with $c$ colors, contains a monochromatic $r$-gon (not necessarily convex) with at most $s$ interior points. Define $λ_r^{\\star}(c)$ to be the smallest integer such that $\\mathsf{M}_r^{\\star}(c,λ_r^{\\star}(c)) < \\infty$. The case of triangles, that is, $r=3$, has been studied in recent years. In this paper, we establish bounds on $λ_r^{\\star}(c)$, for $r \\ge 4$. Specifically, in the bichromatic case, we prove that $λ_r^{\\star}(2)\\leq \\lfloor\\frac{r-2}{2}\\rfloor$, for every $r\\geq3$. For hexagons, we sharpen this bound to $λ_6^{\\star}(2)\\leq1$, that is, every sufficiently large bichromatic planar point set in general position contains a monochromatic hexagon with at most one interior point. Next, for any number of colors $c\\geq3$ and $r\\geq3$, we show that $$(r - 2) \\left \\lfloor \\frac{c-1}{2} \\right \\rfloor \\leq λ_r^\\star(c) \\leq (r-2)(c-2)+\\left\\lfloor(r-2)δ_c\\right\\rfloor, $$ for some constant $δ_c \\in (0, \\frac{1}{2}]$, depending on $c$. The upper bounds combine spiral polygonizations with nested convex-hull arguments and, in the hexagon case, incidence counting in $4$-connected triangulations. The lower bounds follow from colored Horton-set constructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23920",
  "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/0v5y51g7vavtn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23920",
  "fulltext_url": null,
  "id": "0v5y51g7vavtn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23920",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.23920",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0v5y51g7vavtn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.23920",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23920"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Almost Empty Monochromatic Polygons in Planar Point Sets",
  "title_text": "Almost Empty Monochromatic Polygons in Planar Point Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0v5y51g7vavtn",
  "venue": "arXiv",
  "versions": []
}