Erdős Problem #1207
Let P_d(n) be such that in any set of n points in ℝ^d there exist at least P_d(n) many points which do not contain an isosceles triangle. Estimate P_d(n) - in particular, is it true that P_2(n)<n^1-c for some constant c>0?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #1207 as open (status updated 4 April 2026).
- Formal Conjectures marks the Lean statement
erdos_1207as solved.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #1207 on erdosproblems.com | open 2026-04-04 | erdosproblems.com |
| Formal Conjectures: status of `erdos_1207` | research solved | Formal Conjectures |
| Formal Conjectures: research statements in the file | 0 open, 1 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.
Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.
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.
Cite
@misc{cairn-erdos-1207,
title = {Erdős Problem #1207},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-1207}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-1207)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Let be such that in any set of points in there exist at least many points which do not contain an isosceles triangle. Estimate - in particular, is it true that for some constant ?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«1207». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem erdos_1207 : answer(sorry) ↔
∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, (P 2 n : ℝ) < (n : ℝ) ^ (1 - c)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/1207. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
- erdosproblems.com/1207
- [BMP05] Brass, Peter and Moser, William and Pach, János, Research problems in discrete geometry. (2005), xii+499.
- [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
- [PaTa02] Pach, János and Tardos, Gábor, Isosceles triangles determined by a planar point set. Graphs Combin. (2002), 769--779.
Source and licence
Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.