Problem 958

../

claims/: The 2 claim pages of Problem 958, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $A\subset \mathbb{R}^2$ be a finite set of size $n$, and let $\{d_1,\ldots,d_k\}$ be the set of distances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of ordered pairs from $A$ of distance $d$ apart.

Is it true that $k=n-1$ and $\{f(d_i)\}=\{n-1,\ldots,1\}$ if and only if $A$ is a set of equidistant points on a line or a circle?

Statement (corrected). Let $A\subset \mathbb{R}^2$ be a finite set of size $n$, and let $\{d_1,\ldots,d_k\}$ be the set of distances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of unordered pairs from $A$ of distance $d$ apart.

Is it true that $k=n-1$ and $\{f(d_i)\}=\{n-1,\ldots,1\}$ if and only if $A$ is a set of equidistant points on a line or a circle?

Notes. The site's wording counts ordered pairs, and under that count it fails at every $n\ge2$: each unordered pair at distance $d$ gives two ordered pairs, so every multiplicity is even and the profile $\{f(d_i)\}=\{n-1,\ldots,1\}$, which contains $1$, never occurs. The $n$ equally spaced points $0,1,\ldots,n-1$ on a line, which the question names, then have the multiplicities $2(n-1),\ldots,2$, so the "if" direction fails and the answer is no for a reason that has nothing to do with the problem; the smallest instance is $n=2$, two points with the one multiplicity $2$. The change replaces "ordered pairs" by "unordered pairs"; nothing else changes. The evidence is the counting convention of the sources. Erdős's own statement [Er84c, p. 135] conjectures that the multiplicities $u_1,\ldots,u_m$ of the distinct distances among $n$ points cannot be "a permutation of $1,2,\dots,n-1$" unless the points are equidistant on a line or a circle; such multiplicities sum to $\binom n2$, the number of unordered pairs, and equally spaced points on a line achieve the profile only under that count. [CDL25, Section 5, p. 9] states the question for multiplicities whose sum is $\binom n2$. The site's own commentary credits the arc-and-center family of [CDL25] as a further configuration with the profile, which it is only under the unordered count. The defect is the site's: Erdős's text speaks of the multiplicities of the distances and not of ordered pairs. The form follows from these sources, not from the results that settle it. No result concerns the ordered count alone. The page's standing judges the corrected Statement.

Formulation. Erdős [Er84c, p. 135] conjectured the characterization for $n>4$, noting that for $n=4$ the profile is "clearly possible" off lines and circles, by the vertices of an isosceles triangle with the center of its circumscribed circle. He then reported counterexamples to his conjecture for $n=5$, found by Pomerance, and for $n=6$, communicated by L. Berkes, and wrote that he was fairly sure the conjecture holds for sufficiently large $n$, perhaps for all $n>5$. [CDL25] poses the question for all sufficiently large $n$, reporting that Erdős conjectured the characterization for large $n$, and the formal-conjectures statement file at the revision linked below asks it for all sufficiently large $n$ as well. The site's wording asks the question for every $n$, without Erdős's condition. That omission is not corrected: the characterization fails at every $n\ge4$ by the arc-and-center family of [CDL25], and Erdős himself reports failures of his $n>4$ form at $n=5$ and $n=6$, so the failures are not confined to the smallest $n$ and no range removes them. The three questions, for every $n$, for $n>4$ and for all sufficiently large $n$, have the same answer, no, as the Current assessment records.

Status. DISPROVED (LEAN), in the site's label. The site marks the problem disproved, crediting Clemen, Dumitrescu and Liu with a second family of configurations, and flags a Lean proof of a four-point counterexample; both have the profile under the unordered count, so the label describes the corrected Statement. See the accepted claim page and the Lean claim page.

Source. erdosproblems.com/958, accessed 2026-09-04 and 2026-10-07. Cite as: T. F. Bloom, Erdős Problem #958, https://www.erdosproblems.com/958.

References.

• [CDL25] F. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances. arXiv:2505.04283 (2025). Acta Math. Hungar. 177 (2025), 231-245.
• [Er84c] Erdős, P., Some old and new problems in combinatorial geometry. Annals of Discrete Math. 20 (1984), Convexity and graph theory (Jerusalem, 1981), 129-136.

Formalization. Statement in formal-conjectures, which at the revision linked asks the question for all sufficiently large $n$, marks it solved with the answer false crediting [CDL25], names the four-point set below among the small exceptions, and repeats the site's remark on Erdős's conjecture. Boris Alexeev's repository holds a Lean proof, with Aristotle (Harmonic) as formal co-author, that the four points $(0,0)$, $(1,0)$, $(0,1)$, $(0,-1)$ have the profile and lie on no line and no circle; its multiplicities count unordered pairs of distinct points, so it refutes the corrected Statement. It is recorded on its own claim page, has not been built here, and is not native Lean coverage.

Current assessment

Disproved. The corrected Statement asks whether a finite planar set has $k=n-1$ distinct distances with multiplicities $\{n-1,\ldots,1\}$, counted over unordered pairs, if and only if it consists of equally spaced points on a line or on a circle. The answer is no. Clemen, Dumitrescu and Liu, on the accepted claim page, give $n-1$ equally spaced points on a short arc of a unit circle together with the center, which has the profile for every $n\geq4$ and lies on no line or circle; the paper is refereed (Acta Math. Hungar. 177 (2025)) and the site's curator credits it. The standing derives from that claim page. The same family refutes Erdős's conjecture, for $n>4$ [Er84c, p. 135] and for all sufficiently large $n$. The first refutation in print is Erdős's own $n=4$ configuration in [Er84c, p. 135], which he gave as an exception lying outside his conjecture rather than as an answer to the question; it has no claim page of its own, and the four-point set on the Lean claim page is an instance of it.

Contradiction. The site's remark, which the formal-conjectures docstring repeats, says that Erdős conjectured the answer to be no, that is, that other configurations exist. Erdős's own text [Er84c, p. 135] and Section 5 of [CDL25], which calls its family a counterexample to his conjecture, say the opposite. The contradiction is unresolved on the site; this page follows the two sources.

The "if" direction. Equally spaced points on a line have the profile, and so do equally spaced points on a circle whenever the $n-1$ chord lengths they determine are distinct, which holds when the points lie on at most a semicircle; it fails for equally spaced points around the whole circle, since the regular $n$-gon determines only $\lfloor n/2\rfloor$ distinct distances (a remark of this page; Figure 3 of [CDL25] draws the circle case as points on a circular segment). The "if" direction of the question holds with that qualification.

Pending claim. The four-point set $(0,0)$, $(1,0)$, $(0,1)$, $(0,-1)$, found by Aristotle (Harmonic) and proved in Lean in Boris Alexeev's repository, is recorded as a claimed disproof on its own claim page; it has not been built here. It is a right isosceles triangle with its circumcenter, an instance of the $n=4$ example Erdős gave in [Er84c], which the claim page discloses. It refutes the corrected Statement at $n=4$ and leaves Erdős's question for $n>4$, and for all sufficiently large $n$ as the formal-conjectures file reads it, to the family of Clemen, Dumitrescu and Liu. The technical report on ByteDance's Seed-Prover 1.5 (arXiv:2512.17260, 19 December 2025), to which a comment on the site's discussion thread points, lists Problem 958 among fifteen Erdős problems that system solved, adding that these problems are mathematically relatively simple; it releases no proof and no Lean file for the problem, so it asserts a solution with nothing to page and gets no claim page. Search scope, 2026-10-07: the site's problem page and discussion thread, the arXiv and publisher records of [CDL25], [Er84c], the Lean repository and the formal-conjectures file; no forum proof claim, release item or lead names the problem.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• clemen_2025_multiplicities_interpoint_distances
• clemen_2025_multiplicities_interpoint_distances / observation_5_1
• clemen_2025_multiplicities_interpoint_distances / proposition_5_3
• erdos_1984_old_new_problems_combinatorial_geometry
• erdos_1984_old_new_problems_combinatorial_geometry / conjecture_p135