Erdős Problem #3
If A ⊂ ℕ has Σ_n ∈ Afrac 1 n = ∞, then must A contain arbitrarily long arithmetic progressions?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #3 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_3as solved.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #3 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_3` | research solved | Formal Conjectures |
| Formal Conjectures: research statements in the file | 0 open, 6 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-3,
title = {Erdős Problem #3},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-3}},
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-3)
- 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
If has , then must contain arbitrarily long arithmetic progressions?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«3». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ,
(¬ Summable fun a : A ↦ 1 / (a : ℝ)) →
∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength kWhat 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/3. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
- erdosproblems.com/3
- [BlSi20] Bloom, Thomas F. and Sisask, Olof, *Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions*. arXiv:2007.03528 (2020).
- [Go01] Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.
- [GrTa17] Green, Ben and Tao, Terence, *New bounds for Szemerédi's theorem, III: a polylogarithmic bound for *. Mathematika (2017).
- [KeMe23] Kelley, Zander and Meka, Raghu, Strong bounds for 3-progressions. 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) (2023).
- [LSS24] Leng, James, Sah, Ashwin and Sawhney, Mehtaab, *Improved bounds for Szemerédi's theorem*. arXiv:2402.17995 (2024).
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.