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 statement13 of 13

Trivial lower bound: a single-element set is always independent.

For any family A, any singleton {x} is vacuously independent: there are no two distinct elements.

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_501.variants.singleton_independent1 lineExact file

1∀ (A : ℝ → Set ℝ) (x : ℝ), {x}.Pairwise fun x y => x ∉ A y

TextbookStatement only, no proof