Erdős Problem #195 — Monotone arithmetic progressions in permutations of the integers

What is the largest natural number $k$ such that every permutation of $\mathbb Z$ contains a monotone $k$-term arithmetic progression? Equivalently, determine

$$\sup\left\{k\in\mathbb N:\text{ for every bijection }f:\mathbb Z\to\mathbb Z,\ f\text{ contains a }k\text{-term arithmetic progression in monotone order}\right\}.$$

Here “contains in monotone order” means that the terms of the arithmetic progression occur in the order prescribed by the permutation.