Open, the site's label (OPEN). The largest $k$ is $3$ or $4$. Every permutation of $\mathbb{Z}$ contains a monotone three-term progression: its positive terms, in order, form a permutation of the positive integers, and every such permutation contains an increasing three-term progression (Davis, Entringer, Graham and Simmons, Acta Arith. 34 (1977/78), Fact 3; source card), as Adenwalla's introduction notes. Adenwalla's permutation of $\mathbb{Z}$ with no monotone five-term progression gives $k\le4$ (claim page (Adenwalla, 2022), accepted on its refereed publication), improving Geneson's $k\le5$ (claim page (Geneson, 2018), accepted on its refereed publication). Neither result decides whether a monotone four-term progression is forced.