Erdős problem 501

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 $x, y \in X$?