Forbidden arithmetic progressions in permutations of subsets of the integers

TL;DR

This paper advances the understanding of permutations of integers avoiding specific arithmetic progressions, constructing new permutations and establishing density bounds, thereby extending previous results in combinatorial number theory.

Contribution

It constructs permutations avoiding length 6 progressions and improves density bounds for avoiding length 4 progressions, generalizing prior work on forbidden arithmetic progressions.

Findings

Constructed a permutation of integers avoiding length 6 arithmetic progressions.

Proved a lower density bound of 1/2 for avoiding length 4 progressions.

Generalized results to permutations avoiding generalized arithmetic progressions.

Abstract

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). In addition, we generalize several results about forbidden arithmetic progressions to construct permutations avoiding generalized arithmetic progressions.