Erdős Problem 199

References:

• erdosproblems.com/199
• [Ba75] Baumgartner, James E., Partitioning vector spaces. J. Combinatorial Theory Ser. A (1975), 231-233.

theorem Erdos199.erdos_199 : False ↔ ∀ (A : Set ℝ), ThreeAPFree A → ∃ (S : Set ℝ), S.IsAPOfLength ⊤ ∧ S ⊆ Aᶜ

If $A\subset \mathbb{R}$ does not contain a 3-term arithmetic progression then must $\mathbb{R}\backslash A$ contain an infinite arithmetic progression?

Baumgartner [Ba75] answered this in the negative.