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Erdős Problem #623

Combinatorics · posed by Paul Erdős, András Hajnal, 1958 · candidate

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Statement

Let XX be a set of cardinality ℵω\aleph_\omega and ff 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 independent Y⊆XY\subseteq X, i.e. with f(B)∉Yf(B)\not\in Y for all finite B⊂YB\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.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Sungchul Lee, using GPT-5.5 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    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.

    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

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