How to play Thue games

A famous theorem of Thue asserts that there exist arbitrarily long words without repetitions (identical adjacent blocks) over a 3-letter alphabet. In this paper we study a game-theoretic variant of this result, where a word is constructed jointly by two players who alternately append letters to the end of existing word. One of the players (Ann) takes care of avoiding repetitions, while the other one (Ben) tries to force them. We provide an explicit strategy for Ann to avoid non-trivial repetitions over a 9-letter alphabet. This solves an open problem posed by Pegden [13]. We also consider a similar game for overlaps in words (identical overlapping blocks). We give an explicit strategy for Ann to avoid overlaps over a 4-letter alphabet, even if the players are placing letters on arbitrarily chosen positions in constructed word (provided its final length is even). Additionally we prove that the bound of 4 in this result is optimal.