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Level A · Machine-checkable Hard Logic & formalisation P-erdos-623

Erdős Problem #623

Let X be a set of cardinality aleph_ω and f be a function from the finite subsets of X to X such that f(A)not∈ A for all A. Must there exist an infinite Y⊆ X that is independent - that is, for all finite B⊂ Y we have f(B)not∈ Y?

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.

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Cite 
@misc{cairn-erdos-623,
  title        = {Erdős Problem #623},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-623}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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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

Let be a set of cardinality and be a function from the finite subsets of to such that for all . Must there exist an infinite that is independent - that is, for all finite we have ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«623». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_623 : answer(sorry) ↔ ∀ (X : Type u) (hX : #X = ℵ_ ω)
    (f : Finset X → X), (∀ A : Finset X, f A ∉ A) →
    (∃ Y : Set X, Set.Infinite Y ∧ (∀ (B : Finset X), ↑B ⊆ Y → f B ∉ Y))

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/623. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/623

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.