A Generalisation of a Result on Monotone Arithmetic Progressions in Permutations of the Positive Integers

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 $k\ge 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length $4$ with common difference not divisible by $2^k$.