{
  "abstract": "Given a set $S$ of $n$ points in general position in the plane, we define the $C$-set $C(p,q)$ of an oriented edge $(p,q)$ as the set of extreme points of the visibility set $V(p,q)$ facing $(p,q)$. Since every point of $V(p,q)$ that is not in $C(p,q)$ gives rise to an empty convex pentagon having $(p,q)$ as a boundary edge, it is natural to study the properties of $C$-sets in order to bound the number of empty convex pentagons of $S$. We prove tight bounds on the intersection size and union complexity of $C$-sets sharing an endpoint. We also consider the problem of bounding the number of empty pentagons and hexagons in terms of the number of interior and boundary points of $S$. While the hexagon bound is weak, it offers a potential step toward a superlinear lower bound. On the other hand, the two-parameter bound for pentagons, combined with the structural results on $C$-sets, yields a simple proof of an $Ω(n^{3/2})$ bound for the number of pentagons. While this is a significant improvement over previously published bounds, a recent preprint \\cite{Astudillo2026} attains a better bound using completely different methods. Thus, the primary contribution of this paper lies in the structural results about $C$-sets and the two-parameter bounds for empty pentagons and hexagons.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23553",
  "author_total": 1,
  "authors": [
    {
      "name": "Carlos M. Nicolas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/edqbywc87tbhb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23553",
  "fulltext_url": null,
  "id": "edqbywc87tbhb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23553",
  "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",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7222964480",
  "pdf_url": "https://arxiv.org/pdf/2610.23553",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/edqbywc87tbhb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23553",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23553"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Visibility Sets, Empty Pentagons, and Sufficient Conditions for Hexagons",
  "title_text": "Visibility Sets, Empty Pentagons, and Sufficient Conditions for Hexagons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/edqbywc87tbhb",
  "venue": "arXiv",
  "versions": []
}