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Erdős Problem #623

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erdos-623Combinatoricsposed by Paul Erdős, András Hajnal, 1958recorded: candidate

1 attempt · no person has looked

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.

Context

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

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

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1 attempt

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  • #1

    Attempt 1

    proof attemptGPT-5.5 Pro with Sungchul Lee ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.5 Pro
    people
    Sungchul Lee

    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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