Problem detail · source-aware
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Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$. Must there exist an infinite independent $Y\subseteq X$, i.e. with $f(B)\not\in Y$ for all finite $B\subset Y$? Claimed resolution: the positive assertion is equivalent to Koepke's free-subset property, hence independent of ZFC, with consistency strength exactly a measurable cardinal.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
The equivalence to Koepke's free-subset property and the resulting consistency analysis were obtained with the assistance of GPT-5.5 Pro; the author checked the mathematical details.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
An AI screening on the forum found no issues and forum readers concur it would fully resolve the problem in the set-theoretic sense, but there is no independent expert review and erdosproblems.com still lists the problem open.
Correctness: unknown · statement fidelity: unaudited · peer review: none
Resolved (if correct) by an independence result rather than a proof or disproof in ZFC: consistency of the positive answer is equivalent to a measurable cardinal, of the negative to ZFC alone
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.