{
  "abstract": "An equilateral set is a set of points in a metric space whose pairwise distances are all equal. Kusner conjectured that the maximum cardinality of an equilateral set in $\\mathbb{R}^n$ with the $\\ell_p$ metric is $n+1$ for every $1<p<\\infty$. We disprove the conjecture for all $p>4$. In particular, for every such $p$, we define $m=m(p)$ and construct an equilateral set of $8m$ points in $\\mathbb{R}^{8m-2}$. Previously, Swanepoel disproved the conjecture for $1<p<2$, while Ge, Xu, and Zhou showed that it holds for $2\\le p\\le 4$. Combining these prior results with the paper's main theorem resolves Kusner's conjecture for all $1<p<\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14794",
  "author_total": 1,
  "authors": [
    {
      "name": "Nathan Xiong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jw5g8qrg5tsx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14794",
  "fulltext_url": null,
  "id": "7jw5g8qrg5tsx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14794",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213265462",
  "pdf_url": "https://arxiv.org/pdf/2609.14794",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jw5g8qrg5tsx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14794"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Kusner's conjecture is false for $p>4$",
  "title_text": "Kusner's conjecture is false for p>4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7jw5g8qrg5tsx",
  "venue": "arXiv",
  "versions": []
}