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Source. F. C. Clemen, A. Dumitrescu and D. Liu, On multiplicities of interpoint distances, Acta Math. Hungar. 177 (2025), no. 1, 231--245, DOI 10.1007/s10474-025-01562-y; read as arXiv:2505.04283v5 (3 February 2026), whose printed page numbers equal its PDF pages. Section 5 starts on p. 8; Observation 5.1, its proof and Problem 5.2 are on p. 9. The journal version's pagination and labels were not compared.

Statement

Here $a(X)=(a_1(X),\ldots,a_m(X))$ lists the multiplicities of the $m$ distinct distances of $X$ in decreasing order (p. 1).

Observation 5.1 (p. 9). "Let $\gamma$ be a circular arc subtending a center angle $<\pi/3$ on the circle $C$ of unit radius centered at $c$. Let $X$ consist of $c$ together with a set of $n-1$ equidistant points on $\gamma$. Then $a(X)=(n-1,n-2,\ldots,1)$."

The proof (p. 9) notes that the distances among the $n-1$ points on $\gamma$ have multiplicities $1,2,\ldots,n-2$, that the unit distance from the center occurs $n-1$ times, and that $X$ is not contained in any line or circle. That last property needs $n\ge4$: for $n\le3$ the set lies on a line or a circle.

Context. Section 5 (pp. 8--9): $a(X)$ has at most $n-1$ distinct values, since the multiplicities sum to $\binom n2$, and if it has $n-1$ then $a(X)=(n-1,\ldots,1)$; equidistant points on a line or a circle have this profile. Erdős conjectured (Erdős 1984, p. 135, as cited) that for large $n$ no other configurations do, and the paper presents Observation 5.1 as a simple counterexample. It then asks (Problem 5.2, p. 9): "For sufficiently large $n\in\mathbb N$, are the examples in Figure 3 the only point sets with $a(X)=(n-1,n-2,\ldots,1)$? Are these the only ones with pairwise distinct distance multiplicities?" The second question is answered by Proposition 5.3.

Proof pointer

The short verification on p. 9, as summarized above (Figure 3(c)).

Dependencies and read depth

None external. Read depth: claims checked; Observation 5.1, its proof, Problem 5.2 and the Section 5 context were read clause by clause on the page images of pp. 8--9.

Bears on. #958: for every $n\ge4$, a set with the profile $(n-1,\ldots,1)$ (multiplicities counted over unordered pairs) that is not a set of equidistant points on a line or a circle, so the "only if" direction of the characterization fails.