On non-repetitive sequences of arithmetic progressions: : the cases ▫$k \in \{4,5,6,7,8\}$▫
Lužar, Borut ...
ǂA ǂ▫$d$▫-subsequence of a sequence ▫$\varphi = x_1\dots x_n$▫ 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 ▫$k$▫-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 ▫$k \in \set{1,2,3,5}$▫. Here, we present two different proving techniques, and confirm it for all ▫$k$▫, with ▫$2 \le k \le 8$▫.
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