---

Claim. Zoltán Kovács, A note on Erdős's mysterious remark, proves by polynomial elimination in the computer algebra system Giac that a six-point set in the plane in which every triple spans an isosceles triangle is, up to similarity, the vertices of a regular pentagon together with its center, and derives in Section 4 that no seven-point planar set has the property: any two six-point subsets of such a set would both be a centered pentagon and would have to coincide. The largest isosceles subset of $\mathbb{R}^2$ therefore has $6$ points, the instance $d=2$ of Problem 503, giving an algebraic alternative to Kelly's geometric argument of 1947. The isosceles condition for each of the twenty triples is a degree-six polynomial; the full six-point system is computationally infeasible, so the paper eliminates for five points, obtains a 33-point solution set of pentagon-type and square-with-center configurations, and rules out the latter by a second elimination that returns the unit ideal (card). The paper's other result, on the five-point minimizers of the number of distinct distances, concerns Problem 91.

Covers. The instance $d=2$ of the problem, answered $6$, by an independent proof of the result on Kelly's claim page. Nothing is claimed about $d\ge3$.

Acceptance. The paper is refereed: Annals of Mathematics and Artificial Intelligence, published online 26 January 2026, doi:10.1007/s10472-025-09998-2, as its record gives it; the preprint is arXiv:2412.05190, version 1 of 6 December 2024, under the Creative Commons Attribution 4.0 license. The site cites the note as an alternative proof of the planar value while labeling the problem OPEN, so the curator's remark is context for this partial claim and not acceptance evidence.