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Updated Oct 8

Problem 503

claims/: The 6 claim pages of Problem 503, one per claimant's result; the problem's standing derives from them.

Statement. What is the size of the largest $A\subseteq \mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x,y,z$ from $A$ , at least two of the distances $\lvert x-y\rvert,\lvert y-z\rvert,\lvert x-z\rvert$ are equal.

Status. Open, the site's label. The site records the exact values in the plane, $6$ (Kelly), and in space, $8$ (Croft), Blokhuis's upper bound $\binom{d+2}{2}$ and the lower bound $\binom{d+1}{2}+1$ of Alweiss and Weisenberg. The refereed literature determines the value for every $d\le8$ (Ionin, Kido), and Chojecki's note of 2026 derives the value $276$ for $d=22$ and reduces the problem to the two-distance extremal functions. Each determined instance is a partial claim in claims/, the literature's accepted and Chojecki's pending; no claim settles the general question, so the standing stays open.

Source. erdosproblems.com/503, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #503, https://www.erdosproblems.com/503.

References.

[Bl84] Blokhuis, A., Few-distance sets. (1984), iv+70.
[Cr62] Croft, H. T., $9$ -point and $7$ -point configurations in $3$ -space. Proc. London Math. Soc. (3) (1962), 400-424.
[ErKe47] Erdős, Paul and Kelly, L. M., Elementary Problems and Solutions: Solutions: E735. Amer. Math. Monthly (1947), 227-229.
[Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024); Ann. Math. Artif. Intell., published online 26 January 2026, doi:10.1007/s10472-025-09998-2.
[Io09] Y. J. Ionin, Isosceles sets. Electron. J. Combin. 16 (2009), no. 1, Research Paper 141, doi:10.37236/230.
[Ki06] H. Kido, Classification of isosceles eight-point sets in three-dimensional Euclidean space. European J. Combin. 27 (2006), 329–341, doi:10.1016/j.ejc.2005.01.003.
[Ki10] H. Kido, On isosceles sets in the 4-dimensional Euclidean space. Int. J. Combin. 2010, Article ID 803210, doi:10.1155/2010/803210.

Formalization. Statement in formal-conjectures.

Current assessment

The question is the site's formulation, accessed 2026-09-04 (page last edited 28 October 2025): for each $d$ , the largest size $f(d)$ of a set in $\mathbb{R}^d$ in which every three points determine an isosceles triangle. The general question is open; the value is known in the dimensions below.

Determined values. $f(2)=6$ , Kelly's 1947 solution of Monthly problem E735 [ErKe47], with the centered regular pentagon as the unique extremal set (Kelly; an alternative computer-algebra proof is Kovács's [Ko24c], Kovács). $f(3)=8$ , Croft's nine-point theorem [Cr62] with Kelly's eight-point example (Croft); Kido [Ki06] proved the eight-point set unique. $f(4)=11$ , with exactly two extremal sets, Kido [Ki10] (Kido) and Ionin [Io09]. Ionin's Section 5 determines $f(d)$ for every $d\le8$ , $3,6,8,11,17,28,30,45$ , and every extremal set for $d\le7$ (Ionin). $f(22)=276$ follows from Musin's 275-point spherical two-distance set in $\mathbb{R}^{22}$ with its center and Blokhuis's bound, as Corollary 6.4 of Chojecki's note states (Chojecki); that page is pending, the note being unrefereed.

General bounds. Blokhuis's thesis, Theorem 7.2.5, gives $f(d)\le\binom{d+2}{2}$ , with equality only for a two-distance set or a spherical two-distance set with its center (card); the bound is attained for $d=1,2,6,8$ and $22$ . The lower bound $\binom{d+1}{2}$ is Alweiss's, from the vectors $e_i+e_j$ of distinct coordinate vectors in $\mathbb{R}^{d+1}$ , and Weisenberg's thread post of 9 August 2025 adds their centroid to reach $\binom{d+1}{2}+1$ ; the site records both. These bounds settle no instance of the question, so they have no claim pages. Chojecki's identity $f(d)=\max\{g(d),s(d)+1,s(d-1)+3\}$ , with $g$ and $s$ the Euclidean and spherical two-distance maxima, reduces the problem to the two-distance extremal functions; a gap in its first version, raised in the thread on 26 May 2026, was repaired in the revised note of 27 May 2026, and the thread derives from the identity that $f(d)\le g(d)+2$ . Thread derivations get no page.

Formalization. The formal-conjectures file FormalConjectures/ErdosProblems/503.lean, at its commit of 2026-09-18, states erdos_503 as research open with answer(sorry) and marks the variants R2 (the answer $6$ ), R3 (the answer $8$ ), upper_bound and lower_bound as research solved, all without proof and with no formal proof pointer. Chojecki's Lean file proves the reduction's case analysis from the cited results as hypotheses and has not been built here. No formalization enters the standing.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (18)

Problem 1088 Blokhuis's bound for isosceles sets settles the case n = 3 Kelly's six-point maximum for planar isosceles sets Croft's eight-point maximum for isosceles sets in space Ionin's largest isosceles sets in dimensions up to eight Kido's classification of isosceles sets in four dimensions Kovács's computer-algebra proof of the planar case Chojecki's reduction to two-distance extremal functions problems/distance_problems/E0503/claims Distance Problems distance_problems/blokhuis_1984_few_distance_sets Lemma 7.2.4 (p. 47): an edge coloring with connected color classes and no rainbow triangle uses at most two colors Theorem 4.1.1 (p. 26): an s-distance set in E^d or H^d has at most binom(d+s, s) points Theorem 7.2.2 (p. 47): an indecomposable isosceles set is a two-distance set Theorem 7.2.5 (p. 48): an isosceles set in R^d has at most (d+1)(d+2)/2 points distance_problems/erdos_1975_problems_elementary_combinatorial_geometry Section 3 (p. 104): f(n; k), the points in k-space that force n with all distances distinct distance_problems/kovacs_2024_note_erdos_s_mysterious_remark

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