Claims

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1972_01_01_hechler: The site credits Hechler (Bull. Acad. Polon. Sci., 1972) with a negative answer to the first question under CH, a family of countable, null, bounded sets with no infinite independent set; CH holds in Gödel's constructible universe, so ZFC does not prove the positive answer.

1987_06_01_newelski_pawlikowski_seredynski: Newelski, Pawlikowski and Seredyński prove that a set mapping on the reals with closed values of measure below one has an infinite free set, answering the second question of Problem 501 with a free set of size three.

2026_05_29_lee: Lee proves that if Lebesgue measure extends to a countably additive measure on all subsets of the reals, every family of sets of outer measure below one has an infinite independent set; independence relative to a measurable.

2026_08_16_glazer: Glazer proves the first question of Problem 501 independent of ZFC relative to Con(ZFC), by a random-real model over a CH ground and the CH counterexample; the second question's 1987 theorem is formalized beside it.

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