Erdős Problem #501: infinite independent sets for families of small outer measure

For every $x \in \mathbb{R}$ let $A_x \subset \mathbb{R}$ be a bounded set of Lebesgue outer measure $< 1$. Must there be an infinite independent set, that is an infinite $X \subseteq \mathbb{R}$ with $x \notin A_y$ for all distinct $x, y \in X$? Erdős and Hajnal proved that arbitrarily large finite independent sets exist. Hechler showed in 1972 that the answer is no under the continuum hypothesis, so any positive answer had to come from a model where CH fails, and Sungchul Lee later derived one from a real-valued measurable cardinal. The answer is that neither side is provable. Dropping Lee's large cardinal by transferring his argument to the extension of a model of CH by random reals gives a model where the answer is yes; Hechler's construction gives one where it is no. The question is independent of ZFC.

Independent of ZFC, which is why this entry is the first to carry that result rather than proved or disproved. Both directions are formalized: Hechler's 1972 construction gives a model where the answer is no, and adding $\mathfrak{c}^+$ random reals over a model of CH gives one where it is yes. The credit is shared and mostly human. Newelski, Pawlikowski and Seredynski settled the problem's second question in 1987, and it is formalized here without the boundedness hypothesis. Hechler supplied one direction in 1972. Sungchul Lee derived a positive answer from a real-valued measurable cardinal, assisted by GPT-5.5 Pro, and Nat Sothanaphan observed that the two halves together give independence. What Glazer and Sol added is the removal of the large cardinal. erdosproblems.com still lists #501 as open at the time of writing.