Erdős Problem #195

What is the largest \(k\) such that in any permutation of \(\mathbb{Z}\) there must exist a monotone \(k\)-term arithmetic progression \(x_1 < \cdots < x_k\)? Here a permutation of \(\mathbb{Z}\) is a one-sided arrangement \(a_1, a_2, a_3, \ldots\) of the integers, i.e. a bijection \(\mathbb{N} \to \mathbb{Z}\), and a monotone \(k\)-term arithmetic progression is a subsequence \(a_{i_1}, \ldots, a_{i_k}\) with \(i_1 < \cdots < i_k\) forming an increasing or decreasing arithmetic progression.