Erdős Problem 194 #
References:
- erdosproblems.com/194
- [ABJ11] Ardal, H. and Brown, T. and Jungić, V., Chaotic orderings of the rationals and reals. Amer. Math. Monthly (2011), 921-925.
theorem
Erdos194.erdos_194 :
False ↔ ∀ k ≥ 3,
∀ (r : ℝ → ℝ → Prop),
IsStrictTotalOrder ℝ r → ∃ (s : List ℝ), s.IsAPOfLength k ∧ (List.Pairwise r s ∨ List.Pairwise (flip r) s)
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].