On generalized Stanley sequences

Abstract

Let $\mathbb{N}$ denote the set of all nonnegative integers. Let $k\ge 3$ be an integer and $A_0=\{a_1,\dots ,a_t\}$ $(a_1<\dots <a_t)$ 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.