On non-repetitive sequences of arithmetic progressions: The cases k∈{4,5,6,7,8}

Abstract

A d-subsequence of a sequence φ=x1…xn is a subsequence xixi+dxi+2d…, for any positive integer d and any i, 1⩽i⩽n. A k-Thue sequence is a sequence in which every d-subsequence, for 1⩽d⩽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 sequence of arbitrary lengths. So far, the conjecture has been confirmed for k∈{1,2,3,5}. Here, we present two different proving techniques, and confirm it for all k, with 2⩽k⩽8.