A note on Thue games

TL;DR

This paper improves strategies for a two-player word game where one aims to avoid repetitions while the other tries to enforce them, using entropy compression and probabilistic methods to develop explicit avoidance strategies.

Contribution

It presents a new explicit strategy for Ann to avoid repetitions over a 7-letter alphabet, advancing previous bounds using entropy compression and probabilistic techniques.

Findings

Explicit strategy for Ann on 7-letter alphabet

Improved bounds using entropy compression

Application of Lovász Local Lemma in game strategy

Abstract

In this work we improve on a result from~\cite{GryKosZma15}. In particular, we investigate the situation where a word is constructed jointly by two players who alternately append letters to the end of an existing word. One of the players (Ann) tries to avoid (non-trivial) repetitions, while the other one (Ben) tries to enforce them. We show a construction that is closer to the lower bound showed in~\cite{GryKozMic13} using entropy compression, and building on the probabilistic arguments based on a version of the Lov\'asz Local Lemma from~\cite{Peg11}. We provide an explicit strategy for Ann to avoid (non-trivial) repetitions over a $7$-letter alphabet.