Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 655

claims/: The 1 claim page of Problem 655, one per claimant's result; the problem's standing derives from them.

Statement. Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least

(1+c)\frac{n}{2}

distinct distances determined between the $x_i$ , for some constant $c>0$ and all $n$ sufficiently large?

Status. OPEN, the site's label. The site's commentary records Zach Hunter's observation that equally spaced points on a circle disprove the statement as printed and presumes that a general-position hypothesis was intended, and the page's database box flags the original source as ambiguous. The derived standing departs from the label: it is claimed and disproved, because the page shows only the site's Statement and Hunter's regular polygon disproves it; that claim stays pending because it is unrefereed and the corpus has not built the Lean proof that the formal-conjectures catalog records.

Source. erdosproblems.com/655, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #655, https://www.erdosproblems.com/655.

Formalization. Statement in formal-conjectures.

Current assessment

Printed statement. The page shows only the site's Statement, so the standing answers the site's wording, and that wording is false: the regular $n$ -gon satisfies the hypothesis and spans only $\lfloor n/2\rfloor$ distances, as the claim page credited to Hunter records. That claim is unrefereed, the site labels the problem OPEN, and the Lean proof of it that the formal-conjectures catalog records is not built by the corpus, so the derived standing is claimed and disproved. Chojecki's note of 22 April 2026 shows that $\lfloor n/2\rfloor$ is the exact minimum under the hypothesis.
Corrected versions. The site presumes that points in general position, no three on a line and no four on a circle, were meant. That version is open, and the formal-conjectures catalog states it as a separate open variant. Adding only no three on a line, or only convex position, does not repair the statement, since the regular polygon still qualifies. Chojecki traces the $n/2$ scale to Erdős's 1988 question whether, with no four points on a circle and every circle centered at a point holding at most two others, some point sees more than $(1+c)n/2$ distances; the note lists that question as open.

Linked library material

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Linked from (8)

Hunter's regular polygon against the printed statement problems/distance_problems/E0655/claims Distance Problems distance_problems/chojecki_2026_erdos_problem_655_natural_repairs_exact Corollary 3.2 (p. 4): the same exact minima hold in every family of n-point sets containing the regular n-gon Lemma 2.1 (p. 2): under the local condition A_m every point sees at least ceil((n-1)/m) distances Proposition 5.1 (p. 6): with no three points collinear some point sees at least ceil((n-1)/3) distances Theorem 3.1 (p. 3): under the local condition A_2 the minima of D, M and Sigma are floor(n/2), floor(n/2) and n floor(n/2)

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