{
  "abstract": "A finite set of points in $\\mathbb{R}^d$ is isosceles if every three of its points form an isosceles triangle. Erdős asked in 1946 for the largest size $\\mathrm{iso}(d)$ of an isosceles set in $\\mathbb{R}^d$; the answer was known for $d\\le8$. Let $g(d)$ and $M(d)$ be the largest sizes of a two-distance set and of a spherical two-distance set in $\\mathbb{R}^d$. We prove that $\\mathrm{iso}(d)=\\max\\{g(d),M(d)+1,M(d-1)+3\\}$ for every $d\\ge2$, which reduces the problem to these two classical ones. With the recently determined values of $g(d)$ for $9\\le d\\le22$ we obtain $\\mathrm{iso}(d)=\\binom{d+1}{2}+1$ for $9\\le d\\le21$ and $\\mathrm{iso}(22)=276$, and we show that the extremal set is unique: for $d\\le21$ it is the set of midpoints of the edges of a regular simplex together with its centroid, and for $d=22$ it is the McLaughlin configuration together with its centre. For $10\\le d\\le22$ every isosceles set with more than $\\binom d2+3$ points is a two-distance set or a spherical two-distance set together with its centre. The proofs start from Blokhuis' decomposition of isosceles sets into mutually orthogonal two-distance sets. Counting leaves one configuration open: a two-distance set with $\\binom{d+1}{2}$ points in a hyperplane, the largest number allowed by Blokhuis' bound, together with two points that are mirror images of each other. We exclude it by showing that a two-distance set $T$ attaining Blokhuis' bound in $\\mathbb{R}^n$ is unisolvent for quadratic polynomials and that, if $T$ lies on two concentric spheres, then apart from two exceptional cases its two layers are spherical $2$-designs whose sizes are determined by the radii. We also show that the squared distances of such a set are in ratio $(k-1):k$ with $(2k-1)^2\\le n+1$ if $n\\ge9$. Hence there is none for $9\\le n\\le23$, so that $g(23)\\le299$ and $\\mathrm{iso}(23)=\\max\\{g(23),278\\}\\le299$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05421",
  "author_total": 1,
  "authors": [
    {
      "name": "Hyunjun Yi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3td9x16vz7b96.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05421",
  "fulltext_url": null,
  "id": "3td9x16vz7b96",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05421",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3td9x16vz7b96.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Isosceles sets in Euclidean spaces of dimension nine to twenty-two",
  "title_text": "Isosceles sets in Euclidean spaces of dimension nine to twenty-two",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:46Z",
  "url": "https://research.draconitic.com/paper/3td9x16vz7b96",
  "venue": "arXiv",
  "versions": []
}