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Claim. Write $P$ for the positive assertion of the first question of Problem 501: every family $(A_y)_{y\in\mathbb R}$ of bounded sets with $m^*(A_y)<1$ has an infinite independent set. Theorem 1.1: under ZFC together with FMEA, the assertion that Lebesgue measure extends to a countably additive measure on all subsets of $\mathbb R$, every family with $m^*(A_y)<1$ (bounded or not) has an infinite independent set, so $P$ holds. Corollary 1.2: if $\mathrm{ZFC}+\mathrm{FMEA}$ is consistent, equivalently if ZFC with a measurable cardinal is consistent, then $P$ is independent of ZFC. The negative half of the independence is the counterexample under CH that the note attributes to Hechler and writes out in its Appendix A: enumerate $\mathbb R=\{r_\alpha:\alpha<\omega_1\}$ and put $A_{r_\beta}=\{r_\alpha:\alpha<\beta,\ |r_\alpha|\le|r_\beta|+1\}$, a family of countable, null, bounded sets with no infinite independent set.

The claim concerns the first question only; the second question is not treated.

Hypotheses. The positive answer is proved only under FMEA, which fails under CH and is not a theorem of ZFC, and the independence is relative to the consistency of a measurable cardinal, which ZFC cannot prove, not relative to $\mathrm{Con}(\mathrm{ZFC})$ alone. Both hypotheses are removed by Glazer's later claim, which transfers the conclusion to a random-real extension of a model of CH and is the standing-defining result for the first question.

Source. S. Lee, Relative independence of Erdős problem #501, preprint in the author's repository (the links above are pinned to the commit at the head of its main branch on 2026-10-07, the 2026-06-02 commit that added the lean/ folder), two versions: the first dated 2026-05-30 (PDF created 2026-05-29, announced on the problem's discussion thread the same day), the second dated 2026-06-01, which proves its section inequality directly as Lemma 3.1 where the first cited Kunen's theorem as result 543C of Fremlin's Measure Theory, Volume 5, Chapter 54. Not refereed and not on arXiv. Theorem 1.1, Corollary 1.2, Lemma 2.1, Lemma 3.1 and Appendix A of both versions are recorded clause by clause on the source card, the proofs followed and not verified; an author-recorded reconstruction in the Problem 501 research folder is not a review. The note's "Use of AI" section records that the author used OpenAI's GPT-5.5 Pro to explore proof strategies and obtain the main methodological ideas, verified the details independently and takes responsibility for the manuscript.

Acceptance. Reviewed: the curator of erdosproblems.com (T. F. Bloom) adopted the result into the problem text on 2026-09-03, crediting Lee with the positive answer to the first question under an extension of Lebesgue measure to all subsets of $\mathbb R$. The curator is independent of the claimant. A reader's screening of the first version, reported on the discussion thread on 2026-05-29 as finding no issues and as not comprehensive, is not counted as review. Not refereed. The repository's lean/ folder (added 2026-06-02) states fmea_implies_P, fmea_implies_StrongP and ch_implies_not_P; it was not built in this corpus and its statements were not compared with the paper, so it is a link and not evidence.