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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. Every nine points of space therefore contain three points with three distinct distances, and Kelly's eight-point isosceles set shows that eight do not suffice: in the notation of Problem 1088, $f_3(3)=9$. The theorem is recorded as the answer for $d=3$ of the isosceles-set question on [[problems/distance_problems/E0503/claims/1962_01_01_croft|Croft's claim page for Problem 503]].

Covers. The value $f_3(3)=9$, the instance $d=3$, $n=3$.

Depends on. [[problems/discrete_geometry/E1088/claims/1947_04_01_erdos_kelly|The solution of E735]], whose eight-point isosceles set in space supplies the lower bound that makes the nine-point theorem an exact value.

Acceptance. Refereed: Proceedings of the London Mathematical Society, third series, volume 12 (1962), with its corrigendum in volume 13 (1963). The site's remarks credit $f_3(3)=9$ to this paper, but the site labels the problem OPEN, so the remark is not reviewed evidence. The page is dated to the publication year, the record giving no day.