  A permutation of the positive integers avoiding monotone arithmetic
progressions of length $4$ with odd common difference was constructed in
(LeSaulnier and Vijay, 2011). We generalise this result and show that for each
$k\geq 1$, there exists a permutation of the positive integers that avoids
monotone arithmetic progressions of length $4$ with common difference not
divisible by $2^k$.
