Novel structures in Stanley sequences

TL;DR

This paper introduces modular and basic Stanley sequences, characterizes their structure, and extends results to p-free sequences, enabling the construction of sequences with large gaps and generalizing existing theories.

Contribution

It characterizes well-structured Stanley sequences via modular arithmetic and introduces basic sequences, extending the theory to p-free sequences for odd primes.

Findings

Characterization of modular Stanley sequences

Construction of sequences with arbitrarily large gaps

Generalization to p-free sequences for odd primes

Abstract

Given a set of integers with no three in arithmetic progression, we construct a Stanley sequence by adding integers greedily so that no arithmetic progression is formed. This paper offers two main contributions to the theory of Stanley sequences. First, we characterize well-structured Stanley sequences as solutions to constraints in modular arithmetic, defining the modular Stanley sequences. Second, we introduce the basic Stanley sequences, where elements arise as the sums of subsets of a basis sequence, which in the simplest case is the powers of 3. Applications of our results include the construction of Stanley sequences with arbitrarily large gaps between terms, answering a weak version of a problem by Erd\H{o}s et al. Finally, we generalize many results about Stanley sequences to $p$-free sequences, where $p$ is any odd prime.