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Isosceles sets in Euclidean spaces of dimension nine to twenty-two — arXiv:2611.05421

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.05421
License (as archived): CC0-1.0
Archived: 2026-11-05T18:28:37.475Z
SHA-256 of the archived text: 175d9052d998d07951b9a4e7705917618124907b18b85996938e5e17f61b5bde
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How to cite

Isosceles sets in Euclidean spaces of dimension nine to twenty-two — arXiv:2611.05421 — arXiv (en), https://arxiv.org/abs/2611.05421 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2bb515feeb4c (Metraipolis archive dc_2bb515feeb4c, sha256 175d9052d998d07951b9a4e7705917618124907b18b85996938e5e17f61b5bde)

BibTeX

@misc{dc2bb515feeb4c,
  title = {Isosceles sets in Euclidean spaces of dimension nine to twenty-two — arXiv:2611.05421},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.05421},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2bb515feeb4c; sha256 175d9052d998d07951b9a4e7705917618124907b18b85996938e5e17f61b5bde},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.05421. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $d\le 8$, the last cases being due to Ionin. We prove that $\mathrm{iso}(d)=\binom{d+1}{2}+1$ for $9\le d\le 21$ and $\mathrm{iso}(22)=276$, and 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, and the number $\binom d2+3$ is attained. The proof combines Blokhuis' decomposition of isosceles sets into mutually orthogonal two-distance sets, for which we give a short self-contained proof (the key step is a special case of Gallai's theorem on edge colourings without rainbow triangles), with the recent determination of the maximum two-distance sets in $\mathbb{R}^d$ for $9\le d\le22$ and with the known maximum sizes of spherical two-distance sets. In dimension nine this gives only $\mathrm{iso}(9)\le47$, and the exact value $46$ requires an additional argument about eight-dimensional two-distance sets with at least $44$ points. We also show that $\mathrm{iso}(23)=\max\{g(23),278\}$, where $g(23)$ is the largest size of a two-distance set in $\mathbb{R}^{23}$, so that $278\le\mathrm{iso}(23)\le575$.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Metric Geometry (math.MG) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2611.05421 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-05T18:18:34.003Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).