FormalConjectures.ErdosProblems.«966»

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import FormalConjecturesUtil

Erdős Problem 966

References:

erdosproblems.com/966

[Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.

namespace Erdos966

Let $k,r\geq 2$. Does there exist a set $A\subseteq \mathbb{N}$ that contains no non-trivial arithmetic progression of length $k+1$, yet in any $r$-colouring of $A$ there must exist a monochromatic non-trivial arithmetic progression of length $k$?

Erdős [Er75b] reported that 'Spencer has recently shown that such a sequence exists', but gives no reference.

@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos966.lean"]
theorem erdos_966 : answer(True) ↔
    ∀ k r : ℕ, 2 ≤ k → 2 ≤ r → ∃ A : Set ℕ, A.IsAPOfLengthFree (k + 1) ∧
      ∀ coloring : A → Fin r, ContainsMonoAPofLength coloring k := by⊢ True ↔
  ∀ (k r : ℕ),
    2 ≤ k → 2 ≤ r → ∃ A, A.IsAPOfLengthFree (↑k + 1) ∧ ∀ (coloring : ↑A → Fin r), ContainsMonoAPofLength coloring k
  sorryAll goals completed! 🐙

end Erdos966