Erdős (Erdos) Problem #1087: status in 2026

Erdős Problem #1087

Let f(n) be minimal such that every set of n points in ℝ^2 contains at most f(n) many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate f(n) - in particular, is it true that f(n)≤ n^3+o(1)?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-10-17).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1087 as open (status updated 17 October 2025).
Formal Conjectures marks the Lean statement erdos_1087 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1087 on erdosproblems.com	open 2025-10-17	erdosproblems.com
Formal Conjectures: status of `erdos_1087`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 2 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-1087,
  title        = {Erdős Problem #1087},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1087}},
  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

Let $f (n)$ be minimal such that every set of $n$ points in $R^{2}$ contains at most $f (n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f (n)$ - in particular, is it true that $f (n) \leq n^{3 + o (1)}$ ?

Here a set of four points is degenerate if two of its six pairwise distances are equal, and $f (n) \leq n^{3 + o (1)}$ is read as $f (n) ≪ n^{3 + ε}$ for every $ε > 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1087».

theorem erdos_1087 :
    ∀ ε > (0 : ℝ), (fun n : ℕ ↦ (f n : ℝ)) =O[atTop] fun n : ℕ ↦ (n : ℝ) ^ (3 + ε)

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/1087. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1087
[ErPu71] Erdős, P. and Purdy, G., Some extremal problems in geometry. J. Combinatorial Theory Ser. A 10 (1971), 246-252.

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.