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Claim. There is a permutation of $\mathbb{Z}$ with no monotone six-term arithmetic progression, that is, no subsequence $a_{i_1},\ldots,a_{i_6}$, $i_1<\cdots<i_6$, that is an increasing or decreasing six-term progression (Proposition 1 of the arXiv preprint). The permutation is $0\,X_0^*X_1^*\cdots$, where $X_i^*$ arranges $[10^i,10^{i+1})\cup(-10^{i+1},-10^i]$ with no monotone three-term progression. So the largest $k$ of Problem 195 is at most $5$. J. Geneson, Forbidden arithmetic progressions in permutations of subsets of the integers, Discrete Math. 342 (2019), no. 5, 1489–1491, posted as arXiv:1803.06334 on 2018-03-15 and cited as [Ge19] on the problem page (source card).

Covers. The upper bound $k\le5$, improving the bound $k\le6$ that the permutation of $\mathbb{Z}$ without monotone seven-term progressions of Davis, Entringer, Graham and Simmons gives. Superseded by Adenwalla's $k\le4$ on Adenwalla's claim page.

Depends on. No page of this wiki.

Acceptance. Refereed: Discrete Mathematics 342 (2019), no. 5, 1489–1491, doi:10.1016/j.disc.2019.02.004. The site's commentary credits the bound to [Ge19], but the site labels the problem OPEN, so the credit is not listed as reviewed.