Erdős (Erdos) Problem #657: status in 2026

Erdős Problem #657

Is it true that if A⊂ ℝ^2 is a set of n points such that every subset of 3 points determines 3 distinct distances (that is, A has no isosceles triangles, degenerate ones included) then A determines at least f(n)n distinct distances, for some function f with f(n)→ ∞?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #657 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_657 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #657 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_657`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 4 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-657,
  title        = {Erdős Problem #657},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-657}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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

Is it true that if $A \subset R^{2}$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (that is, $A$ has no isosceles triangles, degenerate ones included) then $A$ determines at least $f (n) n$ distinct distances, for some function $f$ with $f (n) \to \infty$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«657».

theorem erdos_657 :
    ∃ f : ℕ → ℝ, Tendsto f atTop atTop ∧
      ∀ n : ℕ, ∀ A : Finset ℝ², A.card = n → (A : Set ℝ²).IsIsoscelesFree →
        f n * (n : ℝ) ≤ (distinctDistances A : ℝ)

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/657. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_657.variants.higher_dim — In [Er73] Erdős attributes the problem, more generally in ℝ^k, to himself and Davies: for every fixed k ≥ 1, is there a function f with f(n)→ ∞ such that every…

References

erdosproblems.com/657
[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
[ErPa90] Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261-269.
[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
[Du08] Dumitrescu, Adrian, On distinct distances and λ-free point sets. Discrete Math. (2008), 6533-6538.

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.