SOLVED
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].
See also [195] and [196].
