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Claim. No nine points of $\mathbb{R}^3$ form an isosceles set, a set in which every three points determine an isosceles triangle. This is the nine-point theorem of H. T. Croft, 9-point and 7-point configurations in 3-space, Proc. London Math. Soc. (3) 12 (1962), 400–424, with a corrigendum in (3) 13 (1963), 384, the second paper link. Together with Kelly's eight-point isosceles set in space it answers the space question of Monthly problem E735, the least number of points that cannot be so arranged, which is $9$: the largest isosceles subset of $\mathbb{R}^3$ has $8$ points, the instance $d=3$ of Problem 503. Ionin's paper of 2009, Section 1, records the theorem in these terms. Kido proved in 2006 that Kelly's eight-point set is the only one up to similarity, and Ionin's Section 5 gives a short proof of that uniqueness.

Covers. The instance $d=3$ of the problem, answered $8$. The matching example is Kelly's, on his claim page.

Depends on. Kelly's claim page, whose eight-point set in space supplies the lower bound that makes the nine-point theorem an exact value.

Acceptance. The paper is refereed: Proceedings of the London Mathematical Society, third series, volume 12 (1962), with its corrigendum in volume 13 (1963), as the zbMATH record gives them. The site records the value $8$ for $d=3$ with this paper as its source while labeling the problem OPEN, so the curator's remark is context for this partial claim and not acceptance evidence. The page is dated to the publication year, the record giving no day.