Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).