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Claim. There is a permutation $a_1,a_2,\ldots$ of $\mathbb{Z}$ with no subsequence $a_{i_1},\ldots,a_{i_5}$, $i_1<\cdots<i_5$, that forms an increasing or decreasing five-term arithmetic progression (Theorem 1), and a doubly infinite permutation of $\mathbb{Z}$ with the same property (Theorem 2). So the largest $k$ of Problem 195 is at most $4$ on either reading its Formulation records. S. Adenwalla, Avoiding monotone arithmetic progressions in permutations of integers, Discrete Math. 347 (2024), no. 11, Paper No. 114183, posted as arXiv:2211.04451 on 2022-11-05 and cited as [Ad22] on the problem page (source card). The permutation concatenates blocks of integers, each arranged with no monotone three-term progression, after affine maps chosen so that no long monotone progression can straddle blocks.

Covers. The upper bound $k\le4$. With the three-term lower bound on the problem page the answer is $3$ or $4$; the claim does not decide whether every permutation of $\mathbb{Z}$ contains a monotone four-term progression. It supersedes the bound $k\le5$ of Geneson's claim page.

Depends on. No page of this wiki.

Acceptance. Refereed: Discrete Mathematics 347 (2024), no. 11, Paper No. 114183, doi:10.1016/j.disc.2024.114183. The site's commentary credits the bound to [Ad22], but the site labels the problem OPEN, so the credit is not listed as reviewed.