A note on Thue games

Abstract

In this work we improve on a result from [1]. In particular, we investigate the situation where a word is constructed jointly by two players who alternately append letters to the same end of the construction. 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 of 6 showed in [2] using entropy compression while building on the probabilistic arguments based on a version of the Lovász Local Lemma from [4]. Our result provides an explicit strategy for Ann to avoid (non-trivial) repetitions over a 7-letter alphabet.