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OP-9100OpenLogic and foundations

Erdős #623 · set theory

Let XX be a set of cardinality ℵω\aleph_\omega and ff be a function from the finite subsets of XX to XX such that f(A)∉Af(A)\not\in A for all AA. Must there exist an infinite Y⊆XY\subseteq X that is independent - that is…

MSC 03 · Mathematical logic and foundations

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let XX be a set of cardinality ℵω\aleph_\omega and ff be a function from the finite subsets of XX to XX such that f(A)∉Af(A)\not\in A for all AA. Must there exist an infinite Y⊆XY\subseteq X that is independent - that is, for all finite B⊂YB\subset Y we have f(B)∉Yf(B)\not\in Y?

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Background

A problem of Erdős and Hajnal [ErHa58], who proved that if ∣X∣<ℵω\lvert X\rvert <\aleph_\omega then the answer is no. Erdős suggests in [Er99] that this problem is 'perhaps undecidable'.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #623 link
  2. [2]Upstream Lean 4 statement for Erdős #623 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 623 link

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