Problem 670

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claims/: The 1 claim page of Problem 670, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A\subseteq \mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?

Formulation. The site reads the question with $d$ fixed and the $o(1)$ term tending to $0$ as $n\to\infty$, possibly at a rate depending on $d$, and this page's standing targets that reading. Erdős [Er97f, p. 6] calls the bound a conjecture independent of the dimension, which also admits a reading uniform in $d$; he adds that the conjecture is settled only on the line, and Ho [Ho26, Remark 9] states that the fixed-dimension question remains open for every $d\ge2$.

Status. Open. The site labels the problem OPEN (page last edited 17 April 2026). The case $d=1$ is the pending partial claim on Erdős's proof on the line.

Source. erdosproblems.com/670, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #670, https://www.erdosproblems.com/670.

References.

• [Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. Source card.
• [Ho26] B. S. Ho, Erdős's diameter conjecture for separated distances fails in high dimensions. arXiv:2604.15305 (2026).

Formalization. None recorded.

Current assessment

The diameter is trivially at least $\binom n2$, since the $\binom n2$ distances are distinct and differ pairwise by at least $1$. Erdős proved the conjecture for $d=1$; that result is the pending partial claim on its claim page, pending because the site labels the problem OPEN and no evidence that the volume was refereed is recorded. For every fixed $d\ge2$ the question is open.

Ho [Ho26] (arXiv:2604.15305, 16 April 2026) constructs, for every prime power $q$, a set of $n=q+1$ points in $\mathbb R^{q^2+q}=\mathbb R^{n^2-n}$ whose pairwise distances differ by at least $1$ and whose diameter is at most $(1-1/\pi^2+o(1))n^2\approx0.8987n^2$. The paper states that GPT-5.4 Pro was used to discover the construction and that Harmonic Aristotle, with some assistance from GPT-5.4 Pro, formalized the proof in Lean 4; the formalization is at https://github.com/boonsuan/erdos670, and this corpus has not built it. The result refutes the reading uniform in $d$. In a fixed dimension $d$ the construction exists only for the finitely many $n$ with $n^2-n\le d$, so it settles no instance of the question this page targets and is not a claim. The literature search behind this account, dated 2026-10-07, covered the site's page and thread, Erdős's 1997 chapter, Ho's paper and the formal-conjectures repository, which has no statement file for 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.

• ho_2026_erdos_s_diameter_conjecture_separated_distances
• erdos_1997_some_unsolved_problems