Erdős Problem 501

For every $x \in \mathbb{R}$ let $A_x \subset \mathbb{R}$ be a bounded set with outer measure $< 1$. Must there exist an infinite independent set, that is, some infinite $X \subseteq \mathbb{R}$ such that $x \notin A_y$ for all $x \neq y \in X$?

If the sets $A_x$ are closed and have measure $< 1$, then must there exist an independent set of size $3$?

Known results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets. Hechler [He72] showed the answer is no assuming the continuum hypothesis.