The site labels the problem NOT DISPROVABLE, a label which composes the two questions' outcomes by the catalog's rule (the strongest status holding of every part). The two questions have different outcomes. Second question: proved, by Newelski–Pawlikowski–Seredyński 1987, Corollary (1): closed sets of measure $<1$ admit an infinite independent set, hence one of size $3$ (refereed). First question: independent of ZFC relative to $\mathrm{Con}(\mathrm{ZFC})$. The negative answer holds under CH, by the countable, null, bounded construction the site attributes to Hechler [He72], written out in both 2026 notes, and CH holds in Gödel's constructible universe, so ZFC does not prove the positive answer if ZFC is consistent (the not-provable side). The positive answer holds after adding $\omega_2$ random reals to any model of CH (E. Glazer, draft rev10 of 2026-08-16, self-published, Theorem 1.1 and Corollary 1.2 [Gla26]), so ZFC does not refute it if ZFC is consistent (the not-disprovable side); earlier, S. Lee proved the positive answer from a full extension of Lebesgue measure [Lee26], which gives independence relative to a measurable cardinal. The reviewed evidence supporting Glazer's accepted claim is the erdosproblems.com editorial adoption of 2026-09-03, which credits Glazer with the independence; the community database change merged 2026-09-18 (teorth/erdosproblems pull request #400, with the maintainers' recorded reasoning) is its context. Neither 2026 result is refereed. The author's public Lean 4 development (github.com/elliotglazer/erdos501) is an unaudited formalization, a link on the claim pages and not acceptance evidence. Taken as the status of the conjunction, the exact statement would be independent (the conjunction is ZFC-equivalent to the first question). The frontmatter lists the two questions as the problem's parts, and each is settled by an accepted claim page recording the result and its acceptance evidence: Glazer's claim settles the first question (independent) and the Newelski–Pawlikowski–Seredyński claim the second (proved); Hechler's claim settles the first question's not-provable side, which Glazer's claim also carries, and Lee's conditional claim settles neither question alone. The standing in the frontmatter derives from them: solved, with the claim value answered, the schema's value when the accepted parts have different outcomes. It departs from NOT DISPROVABLE because the corpus records each question's settled outcome as a part, where the site's label composes the two outcomes into the strongest status that holds of both.