Avoiding monotone arithmetic progressions in permutations of integers (Q6589147)

A permutation, written as a sequence of numbers, avoids monotone arithmetic progressions of length \(k\) if there is no increasing or decreasing subsequence of the permutation that forms a \(k\)-term arithmetic progression. This paper considers finite permutations of the set \(\{1,\ldots,n\}\), and infinite or doubly infinite permutations of all (or all positive) integers. It improves on previously known results on the possibility of avoiding all monotone arithmetic progressions of a given length, sometimes subject to additional conditions on the common difference. \textit{J. Geneson} [ibid. 342, No. 5, 1489--1491 (2019; Zbl 1407.05007)] proved that the integers have a permutation avoiding monotone arithmetic progressions of length \(6\), the authors improve this bound to \(5\) by constructing a new example. \textit{J. A. Davis} et al. [Acta Arith. 34, 81--90 (1977; Zbl 0326.10045)] constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length \(4\). In a related result, the authors construct a doubly infinite permutation of all integers avoiding monotone arithmetic progressions of length \(5\). A permutation of the positive integers that avoided monotone arithmetic progressions of length \(4\) with odd common difference was constructed by \textit{T. D. LeSaulnier} and \textit{S. Vijay} [Discrete Math. 311, No. 2--3, 205--207 (2011; Zbl 1225.05010)]. The author generalizes this result and shows that for each \(k \geq 1\), there is a permutation of the positive integers that avoids monotone arithmetic progressions of length \(4\) with common difference not divisible by \(2^k\). He also specifies the structure of all finite permutations of \(\{1,\ldots,n\}\) that avoid length \(3\) monotone arithmetic progressions mod \(n\) as defined by Davis et al. [loc. cit.], and he provides an explicit construction for a multiplicative result on permutations that avoid length \(k\) monotone arithmetic progressions mod \(n\).