OPEN

What is the smallest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots<x_k$?

Geneson

[Ge19] proved that $k\leq 5$. Adenwalla

[Ad22] proved that $k\leq 4$.

See also [194] and [196].