Nonrepetitive sequences on arithmetic progressions

TL;DR

This paper extends the concept of nonrepetitive sequences to arithmetic progressions, proving the existence of arbitrarily long sequences with bounded symbols where subsequences indexed by certain arithmetic progressions are nonrepetitive.

Contribution

It provides new bounds for the length of sequences with nonrepetitive subsequences on arithmetic progressions, improving previous results using a novel probabilistic technique.

Findings

Existence of arbitrarily long sequences with bounded symbols and nonrepetitive arithmetic progression subsequences.

New upper bounds on the number of symbols needed for such sequences.

Application of a probabilistic method inspired by the Lovász Local Lemma.

Abstract

A sequence $S=s_{1}s_{2}..._{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We prove that for every $k\geqslant 1$ and every $c\geqslant 1$ there exist arbitrarily long sequences over at most $(1+\frac{1}{c})k+18k^{c/c+1}$ symbols whose subsequences indexed by arithmetic progressions with common differences from the set $\{1,2,...,k\}$ are nonrepetitive. This improves a previous bound obtained in \cite{Grytczuk Rainbow}. Our approach is based on a technique introduced recently in \cite{GrytczukKozikMicek}, which was originally inspired by a constructive proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos \cite{MoserTardos}. We also discuss some related problems that can be successfully attacked by this method.