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 statement8 of 13

Erdős–Hajnal (1960): arbitrarily large finite independent sets exist.

For every n : ℕ and every family A : ℝ → Set ℝ of bounded sets with Lebesgue outer measure < 1, there exists a finite independent set of size at least n.

This was proved by Erdős and Hajnal [ErHa60].

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_501.variants.erdosHajnal_finite4 linesExact file

1True ↔2  ∀ (n : ℕ) (A : ℝ → Set ℝ),3    (∀ (x : ℝ), Bornology.IsBounded (A x)) →4      (∀ (x : ℝ), MeasureTheory.volume.toOuterMeasure (A x) < 1) → ∃ X, n ≤ X.card ∧ (↑X).Pairwise fun x y => x ∉ A y

SolvedStatement only, no proof

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