Erdős Problem #1169
Is it true that ω_1^2 not→ (ω_1^2, 3)^2? A problem of Erdős and Hajnal. Hajnal [Ha71] proved this assuming the continuum hypothesis. The source's displayed relation has blue target 3; its introductory k does not occur in that relation. See also erdosproblems.com/592.
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #1169 as not disprovable (status updated 24 January 2026).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #1169 on erdosproblems.com | not disprovable 2026-01-24 | erdosproblems.com |
| Formal Conjectures: research statements in the file | 2 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.
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.
Cite
@misc{cairn-erdos-1169,
title = {Erdős Problem #1169},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-1169}},
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-1169)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
erdos_1169. Is it true that ?
A problem of Erdős and Hajnal. Hajnal [Ha71] proved this assuming the continuum hypothesis. The source's displayed relation has blue target ; its introductory does not occur in that relation. See also erdosproblems.com/592.
finite_blue_target. The finite-blue-target form asks whether for every . The restriction excludes , where the positive relation holds. A colouring witnessing the negative relation with blue target also witnesses this form.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«1169» (2 statements).
theorem erdos_1169 :
¬ OrdinalRamsey ((ω_ 1) ^ (2 : ℕ)) ((ω_ 1) ^ (2 : ℕ)) (3 : Ordinal)theorem finite_blue_target :
∀ k : ℕ, 3 ≤ k →
¬ OrdinalRamsey ((ω_ 1) ^ (2 : ℕ)) ((ω_ 1) ^ (2 : ℕ)) (k : Ordinal)What counts as progress
- A Lean proof of one of the statements above, pinned as the claim's formal statement.
- 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/1169. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
References:
- [Ha71] Hajnal, A. A negative partition relation. Proc. Nat. Acad. Sci. U.S.A. 68 (1971), pp. 142–144.
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.