On Permutations Avoiding Short Progressions

TL;DR

This paper advances bounds on permutations avoiding short arithmetic progressions, explores densities of such subsets, and constructs specific permutations with or without certain APs, contributing to combinatorial number theory.

Contribution

It improves lower bounds on permutations avoiding 3-term APs and establishes new results on the existence and construction of permutations avoiding certain APs.

Findings

Lower bound on permutations avoiding 3-term APs improved

Any permutation of positive integers contains a 3-term AP with odd difference

Constructed permutation avoiding 4-term AP with odd difference

Abstract

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can be permuted to avoid 3-term and 4-term APs. We also show that any permutation of the positive integers must contain a 3-term AP with odd common difference as a subsequence, and construct a permutation of the positive integers that does not contain any 4-term AP with odd common difference.