Thue-like sequences and rainbow arithmetic progressions (Q1856338)

In this paper so-called \(k\)-nonrepetitive colorings of integers are studied having the property that for every \(r\geq 1\) each block of \(kr\) consecutive numbers contains a \(k\)-term rainbow arithmetic progression of difference \(r\). (A subset of integers is called rainbow if no two of its elements are of the same color.) An upper bound for the minimum number of colors in a \(k\)-nonrepetitive coloring is given.