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 $x, y \in X$ ?

Sources

The source's own material, retained exactly.

14 retained statements36c6d0fef71a

FormalConjectures/ErdosProblems/

501.lean

Retained formal statement3 of 13

Two reals form an independent set for the empty family A _ = ∅: neither 0 nor 1 belongs to ∅, so both conditions of pair_independent_iff hold.

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_501.tests.pair_independent_empty1 lineExact file

1{0, 1}.Pairwise fun x y => x ∉ (fun x => ∅) y

TestStatement only, no proof