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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501.lean

Retained formal statement1 of 13

For every $x \in \mathbb{R}$ let $A_x \subset \mathbb{R}$ be a bounded set with outer measure $< 1$ . Must there exist an infinite independent set, that is, some infinite $X \subseteq \mathbb{R}$ such that $x \notin A_y$ for all $x \neq y \in X$ ?

If the sets $A_x$ are closed and have measure $< 1$ , then must there exist an independent set of size $3$ ?

Known results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets. Hechler [He72] showed the answer is no assuming the continuum hypothesis.

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_5014 linesExact file

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

OpenStatement only, no proof

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