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 $x, y \in X$?

Gladysz (1962) [Gl62]: independent set of size 2 in the closed case.

If all the sets A x are closed with Lebesgue measure < 1, then there exist two distinct reals x y such that x ∉ A y and y ∉ A x.

This is a weaker result proved by Gladysz before the full Newelski–Pawlikowski– Seredyński theorem [NPS87].