Nonrepetitive games

TL;DR

This paper explores game-theoretic versions of nonrepetitive sequences, demonstrating strategies for players to build arbitrarily long sequences avoiding repetitions over small symbol sets, using advanced probabilistic and algorithmic techniques.

Contribution

The authors introduce new strategies for nonrepetitive games, reducing the number of symbols needed from 37 to 6, and analyze erase-repetition games over 8 symbols, employing entropy compression methods.

Findings

First player can build arbitrarily long nonrepetitive sequences over 6 symbols.

In erase-repetition games, the first player can do so over 8 symbols.

Techniques are based on a new algorithmic proof of the Lovász Local Lemma.

Abstract

(Note. The results of this manuscript has been merged and published with another paper of the same authors: A new approach to nonrepetitve sequences.) A repetition of size $h$ ($h\geqslant1$) in a given sequence is a subsequence of consecutive terms of the form: $xx=x_1... x_hx_1... x_h$. A sequence is nonrepetitive if it does not contain a repetition of any size. The remarkable construction of Thue asserts that 3 different symbols are enough to build an arbitrarily long nonrepetitive sequence. We consider game-theoretic versions of results on nonrepetitive sequences. A 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 $>1$. Our techniques allow to reduce 37 to 6. Another game we consider is an 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. Our approach is inspired by a new algorithmic proof of the Lov\'asz Local Lemma due to Moser and Tardos and previous work of Moser (his so called entropy compression argument).