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Claim. The full manuscript gives an extremal formula for $f(n)$, the largest size of a subset of $\{1,\ldots,n\}$ in which no element divides two other distinct elements, and proves that the limiting density $c_*=\lim_{n\to\infty}f(n)/n$ is transcendental, which implies the irrationality Erdős asked about; as posted, the claim answers both questions of the problem, so its scope is full and its value answered. The thread post of 23 September 2026 by the user DavidTurturean, which also acknowledges the Conjectures.io Lean-verified solution of the irrationality question under the username JenW1N and its priority for that result, presents the manuscript as an independently obtained stronger result and links it as a read-only shared document. The post also links a standalone transcendence addendum to the Conjectures.io result, which starts from the density formula in that published Lean proof and derives the transcendence from bounds for linear forms in logarithms and the Subspace Theorem. The post says the transcendence argument was elicited in a single response from GPT-6-Astra Pro. This page records the post's assertions; the linked manuscripts are not held in the library.

Depends on. The standalone addendum's route rests on the density formula of the accepted [[problems/integer_sequences/E1062/claims/2026_09_21_jenw1n|Conjectures.io claim]]; the full manuscript is asserted to be independent of it.

Standing. No reply, review or acceptance appears in the thread as of 2026-10-07, and the manuscripts are unrefereed and unregistered, so the claim is claimed. The irrationality itself is settled on the accepted [[problems/integer_sequences/E1062/claims/2026_09_21_jenw1n|Conjectures.io claim]].