---

Claim. The answer to Problem 958 is no, against Erdős's conjecture. Felix Christian Clemen, Adrian Dumitrescu and Dingyuan Liu, On multiplicities of interpoint distances, Acta Math. Hungar. 177 (2025), 231--245; posted as arXiv:2505.04283 on 7 May 2025. Among their results on how often distances repeat in planar sets, the authors observe (Observation 5.1) that a second family has the profile of the question: take $n-1$ equally spaced points on an arc of a circle of radius $1$ subtending a center angle less than $\pi/3$, together with the center. The center is at distance $1$ from each of the $n-1$ arc points, and the arc points, at angular step $\alpha$, determine the chords $2\sin(j\alpha/2)$ for $j=1,\ldots,n-2$, the chord with index $j$ occurring $n-1-j$ times; the angle condition keeps every chord below $1$, so these are $k=n-1$ distinct distances with multiplicities $n-1,n-2,\ldots,1$. For $n\geq4$ the set lies on no line and on no circle, since three of its points fix the circle and the center is not on it. So the "only if" direction of the question fails for every $n\geq4$: the profile $k=n-1$ with $\{f(d_i)\}=\{n-1,\ldots,1\}$ does not force equally spaced points on a line or a circle. The paper counts multiplicities over unordered pairs, as the corrected Statement of the problem page does. Section 5 of the paper poses the question for all sufficiently large $n$, states that Erdős conjectured that no configurations other than the line and the circle exist for large $n$, and presents the family as a counterexample to that conjecture; Erdős's own text, cited on the problem page as [Er84c, p. 135], conjectures the characterization for $n>4$ and then for all sufficiently large $n$, after recording the exceptions at $n=4$, $5$ and $6$. The site's remark that Erdős conjectured the answer to be no contradicts both sources, which the problem page records as an unresolved contradiction. The family refutes the corrected Statement at every $n\geq4$, and with it Erdős's questions for $n>4$ and for all sufficiently large $n$. The paper is carded at clemen_2025_multiplicities_interpoint_distances.

Acceptance. The result is refereed: the paper appeared in Acta Mathematica Hungarica. The site's curator, Thomas Bloom, marks the problem DISPROVED (LEAN) and credits Clemen, Dumitrescu and Liu with the configuration on the problem page; the curator neither wrote nor submitted the result. This corpus has not reviewed the paper, and no such review is needed for the standing recorded here; the chord computation above is a reading aid. A separate four-point counterexample, found by Aristotle and proved in Lean, is recorded on its own claim page; the site's label carries the Lean marker for it, and it is not a formalization of this result.