On generalized Stanley sequences

Abstract

Let ${\mathbb{N}}$ denote the set of all nonnegative integers. Let ${k \ge 3}$ be an integer and ${A_{0} = \{a_{1}, \dots, a_{t}\} (a_{1} < \cdots < a_{t})}$ be a nonnegative set which does not contain an arithmetic progression of length k. We denote ${A = \{a_{1}, a_{2}, \ldots{}\}}$ defined by the following greedy algorithm: if ${l \ge t}$ and ${a_{1}, \dots{}, a_{l}}$ have already been defined, then ${a_{l+1}}$ is the smallest integer ${a > a_{l}}$ such that ${\{a_{1}, \dots, a_{l}\} \cup \{a\}}$ also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A₀. We prove some results about various generalizations of the Stanley sequence.