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Erdős problem 194

Let k≥3k\geq 3. Must any ordering of R\mathbb{R} contain a monotone kk-term arithmetic progression, that is, some x1<⋯<xkx_1 <\cdots < x_k which forms an increasing or decreasing kk-term arithmetic progression?

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FormalConjectures/ErdosProblems/

194.lean

Retained formal statement1 of 1

Let k≥3k\geq 3. Must any ordering of R\mathbb{R} contain a monotone kk-term arithmetic progression, that is, some x1<⋯<xkx_1 <\cdots < x_k which forms an increasing or decreasing kk-term arithmetic progression?

The answer is no, even for k=3k=3, as shown by Ardal, Brown, and Jungić [ABJ11]. -

FormalConjectures/ErdosProblems/194.leanErdos194.erdos_1944 linesExact file
False ↔  ∀ k ≥ 3,    ∀ (r : ℝ → ℝ → Prop),      IsStrictTotalOrder ℝ r → ∃ s, s.IsAPOfLength k ∧ (List.Pairwise r s ∨ List.Pairwise (flip r) s)
SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

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