Let $k\geq 3$. Must any ordering of $\mathbb{R}$ contain a monotone $k$-term arithmetic progression,
that is, some $x_1 <\cdots < x_k$ which forms an increasing or decreasing $k$-term arithmetic
progression?
The answer is no, even for $k=3$, as shown by Ardal, Brown, and Jungić [ABJ11].
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theorem erdos_194 :
answer(False) ↔ ∀ k ≥ 3, ∀ r : ℝ → ℝ → Prop, IsStrictTotalOrder ℝ r →
∃ s : List ℝ, s.IsAPOfLength k ∧ (s.Pairwise r ∨ s.Pairwise (flip r))
