Skip to content

Erdős problem 501

Start work

For every x∈Rx \in \mathbb{R} let Ax⊂RA_x \subset \mathbb{R} be a bounded set of Lebesgue outer measure <1< 1. Must there be an infinite independent set, that is an infinite X⊆RX \subseteq \mathbb{R} with x∉Ayx \notin A_y for all distinct x,y∈Xx, y \in X?

Sources

The source's own material, retained exactly.

14 retained statements36c6d0fef71a

Open selected source

FormalConjectures/ErdosProblems/

501.lean

Retained formal statement11 of 13

Newelski–Pawlikowski–Seredyński (1987) [NPS87]: infinite independent set in the closed case.

If all the sets A x are closed with Lebesgue measure < 1, then there is an infinite independent set. This gives a strong affirmative answer to the second question of Problem 501.

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_501.variants.newelski_pawlikowski_seredynski4 linesExact file
True ↔  ∀ (A : ℝ → Set ℝ),    (∀ (x : ℝ), IsClosed (A x)) →      (∀ (x : ℝ), MeasureTheory.volume (A x) < 1) → ∃ X, X.Infinite ∧ X.Pairwise fun x y => x ∉ A y
SolvedStatement only, no proof