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

Abstract

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 \(2^k\).