Problem 501

Erdős Problem #501: declared status 'open'. Formalized: yes. 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 $x\not\in A_y$ 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 $3$? Prize: no. Tags: combinatorics, set theory.