On generalized Stanley sequences (1710.01939v1)

Abstract: Let $\mathbb{N}$ denote the set of all nonnegative integers. Let $k\ge 3$ be an integer and $A_{0} = {a_{1}, \dots{}, a_{t}}$ $(a_{1} < \ldots< a_{t})$ be a nonnegative set which does not contain an arithmetic progression of length $k$. We denote $A = {a_{1}, a_{2}, \dots{}}$ 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 $A_{0}$. In this paper, we prove some results about various generalizations of the Stanley sequence.