{
  "abstract": "Let $p(n)$ be the minimum cardinality of a finite planar point set in which each point has exactly $n$ unit-distance neighbours. Harborth proved $p(5)=18$ in work presented in 1985 and published in 1987, using a detailed case analysis. We revisit the proof using convex-hull arguments, incidence counts, and elementary properties of intersections of unit circles. These tools give an alternative treatment of configurations with $14$ and $16$ points and organize the final case distinction by a common vector-sum structure. We also treat a configuration not explicitly covered by the case analysis in the online PDF copy of Harborth's paper available to us, and give an explicit justification of the six-point classification used in the argument. No result subsequent to Harborth's proof is required.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40725",
  "author_total": 1,
  "authors": [
    {
      "name": "Mike Winkler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5se2eeancydtt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40725",
  "fulltext_url": null,
  "id": "5se2eeancydtt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40725",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40725",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5se2eeancydtt.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40725",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40725"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Harborth's theorem p(5)=18 revisited",
  "title_text": "Harborth's theorem p(5)=18 revisited",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/5se2eeancydtt",
  "venue": "arXiv",
  "versions": []
}