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Claim. If ZFC plus a measurable cardinal is consistent, then so is ZFC together with the positive answer to Problem 623: there is a model in which every $f$ from the finite subsets of a set $X$ of size $\aleph_\omega$ to $X$ with $f(A)\notin A$ has an infinite independent set. The claimant's summary describes the proof as a forcing argument built around a structure $S$, and adds a conditional statement: if a set the write-up calls $\mathrm{Good}$ has the full size of $X$, the positive answer holds outright in ZFC. The argument is recorded from the claimant's summary and has not been reconstructed in this corpus; the write-up's hypothesis on $\mathrm{Good}$ is recorded as the claimant states it.

Submission note. Posted to erdosproblems.com as a proof claim by Casey Crawford (account kauauicel) on 21 August 2026, giving "Claude Sonnet 5 High, ChatGPT 5.6 Luna (Thinking)" as the AI used:

This partial proof, which, I will admit, is similar to a prior proof by Sungchul Lee in June, but was done independently, was created by myself mostly as a test at how good I am at set theory. I have proofread the proof, and understand its complexities in its entirety. If it is correct, this proof shows that Erdos Problem 623 is consistent in ZFC relative to a measurable cardinal via forcing, via the construction of a strucutre S. It additionally proves that if |Good| = |X|, that Erdos Problem 623 would therefore be true in ZFC. Notes: Read Sungchul Lee's proof too, he seems a lot better than I am at this. This was made a 14-year old trans girl who loves set theory, and his proof, if true, is more powerful than mine.

Covers. The consistency of a positive answer relative to a measurable cardinal, that is, that ZFC cannot refute the positive answer unless ZFC with a measurable cardinal is inconsistent. It does not address the consistency of a negative answer, so it does not by itself give independence, and it does not claim that the answer is a theorem of ZFC except under the hypothesis on $\mathrm{Good}$ above.

Standing. The claimant is Casey Crawford, who posted the claim on the site's proof-claims tab on 2026-08-21 under the forum name kauauicel as a partial proof, naming the systems Claude Sonnet 5 High and ChatGPT 5.6 Luna (Thinking). The claimant's note says the argument is similar to, but was found independently of, the earlier proof of Lee's independence result, which it calls the stronger of the two. The one comment under the claim, posted by the claimant the same day, says that the claimant understands the mechanism of most, if not all, of the proof, including Koepke's 1984 paper, and that this is the claimant's first paper; it is not a review. The site's label is OPEN, no one has reviewed or refereed the result, and nothing was built here, so the claim stays claimed.