Erdős Problem #598
Erdős Problem 598: Let m be an infinite cardinal and κ be the successor cardinal of 2^aleph_0. Can one colour the countable subsets of m using κ many colours so that every X ⊆ m with |X| = κ contains subsets of all possible colours?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #598 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_598as open.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #598 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_598` | research open | Formal Conjectures |
| Formal Conjectures: research statements in the file | 1 open, 0 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.
Cite
@misc{cairn-erdos-598,
title = {Erdős Problem #598},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-598}},
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-598)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
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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.
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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
Erdős Problem 598: Let be an infinite cardinal and be the successor cardinal of . Can one colour the countable subsets of using many colours so that every with contains subsets of all possible colours?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«598».
theorem erdos_598 :
∀ (m : Type*) [Infinite m],
∃ c : { s : Set m // s.Countable } → κ.out,
∀ X : Set m, #X = κ →
c '' { s : { sub : Set m // sub.Countable } | s.1 ⊆ X } = Set.univWhat 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/598. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
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.