Thue-like Sequences and Rainbow Arithmetic Progressions

A sequence $u=u_{1}u_{2}...u_{n}$ is said to be nonrepetitive if no two adjacent blocks of $u$ are exactly the same. For instance, the sequence $a{\bf bcbc}ba$ contains a repetition $bcbc$, while $abcacbabcbac$ is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set $\{a,b,c\}$. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this paper we consider a stronger property defined as follows. Let $k\geq 2$ be a fixed integer and let $C$ denote a set of colors (or symbols). A coloring $f:{\bf N}\rightarrow C$ of positive integers is said to be $k$-nonrepetitive if for every $r\geq 1$ each segment of $kr$ consecutive numbers contains a $k$-term rainbow arithmetic progression of difference $r$. In particular, among any $k$ consecutive blocks of the sequence $f=f(1)f(2)f(3)...$ no two are identical. By an application of the Lovász Local Lemma we show that the minimum number of colors in a $k$-nonrepetitive coloring is at most $2^{-1}e^{k(2k-1)/(k-1)^{2}}k^{2}(k-1)+1$. Clearly at least $k+1$ colors are needed but whether $O(k)$ suffices remains open. This and other types of nonrepetitiveness can be studied on other structures like graphs, lattices, Euclidean spaces, etc., as well. Unlike for the classical Thue sequences, in most of these situations non-constructive arguments seem to be unavoidable. A few of a range of open problems appearing in this area are presented at the end of the paper.