FormalConjectures.ErdosProblems.«199»

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import FormalConjecturesUtil

Erdős Problem 199

References:

erdosproblems.com/199

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


namespace Erdos199

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.

@[category research solved, AMS 5,
  formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos199.lean"]
theorem declaration uses 'sorry'erdos_199 : answer(False) ↔
    ∀ A : Set ℝ, ThreeAPFree A →
      ∃ S : Set ℝ, S.IsAPOfLength ⊤ ∧ S ⊆ Aᶜ := by⊢ False ↔ ∀ (A : Set ℝ), ThreeAPFree A → ∃ S, S.IsAPOfLength ⊤ ∧ S ⊆ Aᶜ
  sorryAll goals completed! 🐙


end Erdos199