Erdős problem 501 · problems.science

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 $x, y \in X$ ?

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FormalConjectures/ErdosProblems/

501.lean

Retained formal statement7 of 13

Closed sets case: existence of an independent set of size 3.

If the sets A x are closed with Lebesgue measure < 1, must there exist an independent set of size 3?

This is implied by the stronger theorem of Newelski–Pawlikowski–Seredyński [NPS87] below; Gladysz [Gl62] earlier proved the existence of an independent set of size 2.

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_501.variants.closed_size34 linesExact file

1True ↔2  ∀ (A : ℝ → Set ℝ),3    (∀ (x : ℝ), IsClosed (A x)) →4      (∀ (x : ℝ), MeasureTheory.volume (A x) < 1) → ∃ X, 3 ≤ X.ncard ∧ X.Pairwise fun x y => x ∉ A y

SolvedStatement only, no proof

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