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Claim. Call a finite set isosceles when every three of its points form an isosceles triangle. Kelly's solution of Monthly problem E735, which Erdős had posed in 1946, proves that no isosceles set in the plane has seven points and that, up to similarity, the only isosceles set of six points in the plane is the set of vertices of a regular pentagon together with its center. The largest isosceles subset of $\mathbb{R}^2$ therefore has $6$ points, which answers the instance $d=2$ of Problem 503. The solution also exhibits an isosceles set of eight points in $\mathbb{R}^3$, the lower bound that Croft's theorem later matched. Ionin's paper of 2009, Section 1, describes the solution in these terms, and Kovács's note of 2024 re-proves the planar classification by computer algebra (card).

Covers. The instance $d=2$ of the problem, answered $6$, with the six-point set unique up to similarity; and the lower bound $8$ for $d=3$, whose matching upper bound is on Croft's claim page. Nothing is claimed about $d\ge3$ beyond that example.

Acceptance. The solution is refereed: P. Erdős and L. M. Kelly, E735, Amer. Math. Monthly 54 (1947), no. 4, 227–229, in the journal's problems section, Erdős as proposer and Kelly as solver. The site records the value $6$ for $d=2$ with this solution 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 issue, April 1947.