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 statement9 of 13

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].

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_501.variants.gladysz_size24 linesExact file

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

SolvedStatement only, no proof

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