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–Hajnal (1960): arbitrarily large finite independent sets exist.

For every n : ℕ and every family A : ℝ → Set ℝ of bounded sets with Lebesgue outer measure < 1, there exists a finite independent set of size at least n.

This was proved by Erdős and Hajnal [ErHa60].