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 $x, y \in X$ ?

Sources

Browse retained paths and inspect the exact material available for this Problem.

14 retained statements2415f78e850a

Open selected source

FormalConjectures/ErdosProblems/

501.lean

Retained formal statement10 of 13

Hechler (1972) [He72]: the answer to the main question is NO, assuming the continuum hypothesis.

Assuming CH (ℵ₁ = 𝔠), there exists a family A : ℝ → Set ℝ of bounded sets with Lebesgue outer measure < 1 for which no infinite independent set exists.

FormalConjectures/ErdosProblems/501.leanErdos501.erdos_501.variants.hechler_CH5 linesExact file

1True ↔2  Cardinal.aleph 1 = Cardinal.continuum →3    ∃ A,4      (∀ (x : ℝ), Bornology.IsBounded (A x)) ∧5        (∀ (x : ℝ), MeasureTheory.volume.toOuterMeasure (A x) < 1) ∧ ¬∃ X, X.Infinite ∧ X.Pairwise fun x y => x ∉ A y

SolvedStatement only, no proof

Sources

Search problems.science