On non-repetitive sequences of arithmetic progressions:the cases $k\in \{4,5,6,7,8\}$

Abstract

A $d$-subsequence of a sequence $\phi =x_1\dots x_n$ is a subsequence $x_i{x}_{i+d}{x}_{i+2d}\dots$, for any positive integer $d$ and any $i$, $1\le i\le n$. A \textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence, for $1\le d\le k$, is non-repetitive, i.e. it contains no consecutive equal subsequences. In 2002, Grytczuk proposed a conjecture that for any $k$, $k+2$ symbols are enough to construct a $k$-Thue sequences of arbitrary lengths. So far, the conjecture has been confirmed for $k\in \{1,2,3,5\}$. Here, we present two different proving techniques, and confirm it for all $k$, with $2\le k\le 8$.