Problem 503

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.

Source: erdosproblems.com/503

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

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.

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.

Problem 503

Status

Proof claims

Discussion