Given a permutation, $P$, of a set of integers, a $k$ term monotone arithmetic progression is a $k$ term increasing or decreasing subsequence of $P$ that form a $k$ term arithmetic progression. If you have a permutation of an infinite subset of the integers and it has a finite number of $3$ term monotone arithmetic progressions, can the permutation necessarily be turned into one free of monotone arithmetic progressions by moving the position of a finite number of integers in the permutation?
Edit: Assume you have a permutation of the integers (or positive integers) that has a finite number of $k$ term monotone arithmetic progressions. Let these progressions have common differences $d_1,d_2,\ldots,d_n$. Then there exists some integer $m$ that is coprime to every common difference, that is $\gcd(m,d_i)=1$ for $i=1,\ldots,n$. Then take the subsequence of the multiples of $m$ in your permutation. Any $k$ term monotone arithmetic progression in this subsequence would have common difference divisible by $m$, so the subsequence must be free of $k$ term monotone arithmetic progressions. Divide each term in the subsequence by $m$ and you get a permutation of the integers (or positive integers) that is free of $k$ term monotone arithmetic progressions.