Erdős Problem #598 — open problem in logic & formalisation

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?

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.

Start working on it Submit a claim Follow

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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Erdős Problem 598: Let $m$ be an infinite cardinal and $κ$ be the successor cardinal of $2^{ℵ_{0}}$ . Can one colour the countable subsets of $m$ using $κ$ many colours so that every $X \subseteq m$ with $∣ X ∣ = κ$ contains subsets of all possible colours?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«598». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_598 : answer(sorry) ↔
    ∀ (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.univ

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/598. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/598

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.