Erdős (Erdos) Problem #670: status in 2026

Erdős Problem #670

Let A⊆ 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? The quantifiers in the source are ambiguous.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-04-16).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #670 as open (status updated 16 April 2026).
Formal Conjectures marks the Lean statement erdos_670 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #670 on erdosproblems.com	open 2026-04-16	erdosproblems.com
Formal Conjectures: status of `erdos_670`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-670,
  title        = {Erdős Problem #670},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-670}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

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

The quantifiers in the source are ambiguous. Following the remarks on erdosproblems.com, this is the regime where $d$ is fixed and the $o (1)$ term tends to $0$ as $n \to \infty$ , at a rate which may depend on $d$ . The case $d = 1$ was proved by Erdős (Erdos670.erdos_670.variants.dim_one), so the question is open for $d \geq 2$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«670».

theorem erdos_670 :
    ∀ d : ℕ, 1 ≤ d → ∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
      ∀ A : Finset (EuclideanSpace ℝ (Fin d)), A.card = n → HasSeparatedDistances A →
        (1 - ε) * (n : ℝ) ^ 2 ≤ Metric.diam (A : Set (EuclideanSpace ℝ (Fin d)))

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/670. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/670
[Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.
[Ho26] Ho, B. S., *Erdős's diameter conjecture for separated distances fails in high dimensions*. arXiv:2604.15305 (2026).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.