New approach to nonrepetitive sequences

Abstract

A sequence is nonrepetitive if it does not contain two adjacent identical blocks. The remarkable construction of Thue asserts that three symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of three‐element sets L1,…,Ln there exists a nonrepetitive sequence s1,…,sn with si∈Li. We propose a new non‐constructive way to build long nonrepetitive sequences and provide an elementary proof that sets of size 4 suffice confirming the best known bound. The simple double counting in the heart of the argument is inspired by the recent algorithmic proof of the Lovász local lemma due to Moser and Tardos. Furthermore we apply this approach and present game‐theoretic type results on nonrepetitive sequences. Nonrepetitive game is played by two players who pick, one by one, consecutive terms of a sequence over a given set of symbols. The first player tries to avoid repetitions, while the second player, in contrast, wants to create them. Of course, by simple imitation, the second player can force lots of repetitions of size 1. However, as proved by Pegden, there is a strategy for the first player to build an arbitrarily long sequence over 37 symbols with no repetitions of size greater than 1. Our techniques allow to reduce 37–6. Another game we consider is the erase‐repetition game. Here, whenever a repetition occurs, the repeated block is immediately erased and the next player to move continues the play. We prove that there is a strategy for the first player to build an arbitrarily long nonrepetitive sequence over 8 symbols. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

A note on Thue games

“…For more details see [2]. The game is played between two players which add letters one after the other (starting from scratch) to the same end of the existing sequence.…”

“…There have been quite a number of results regarding the winning strategies that Ann should employ depending on the possible length of the repeating sequence [2,4,1], most of which proved only the existence of such a strategy. However, when considering the case of sequences of length 2 or longer (nontrivial repetitions), which is the basic case given the fact that repetitions of length 2 are already created by Ben whenever he copies the symbol that Ann has previously added, deterministic approaches of such an algorithm have been explicitly provided for an alphabet of size 37 in [4] and later on improved to an alphabet of size 9 in [1].…”

Witness Trees in the Moser–Tardos Algorithmic Lovász Local Lemma and Penrose Trees in the Hard-Core Lattice Gas

“…the collection of bad events A = {A x } x∈X depending on a finite number of independent random variables ν ∈ V, it is possible to construct different algorithms which could be more efficient and stops even for a set of probabilities p for which the Moser Tardos algorithm doesn't stop. Along these directions we would like to cite some interesting results obtained in [11] and [8].…”

Pattern avoidance in partial words over a ternary alphabet

“…Proof. The proof follows the general framework of Moser-Tardosz algorithmisation of Lovasz local lemma [7] adjusted for application to sequences by Grytczuk, Kozik and Micek [4]. We will refer to this method as entropy compression.…”