Ann wins the nonrepetitive game over four letters and the erase-repetition game over six letters

TL;DR

This paper improves bounds on the alphabet size needed for Ann to win two combinatorial word games, using a new counting method that simplifies previous complex probabilistic approaches.

Contribution

It reduces the known alphabet size bounds for winning strategies in both the nonrepetitive and erase-repetition games by applying a novel counting argument.

Findings

Ann wins the nonrepetitive game with an alphabet of size 4 or more.

Ann wins the erase-repetition game with an alphabet of size 6 or more.

The new counting method simplifies the analysis compared to entropy compression and Lovász Local Lemma approaches.

Abstract

We consider two games between two players Ann and Ben who build a word together by adding alternatively a letter at the end of the shared word. In the nonrepetitive game, Ben wins the game if he can create a square of length at least $4$, and Ann wins if she can build an arbitrarily long word before that. In the erase-repetition game, whenever a square occurs the second part of the square is erased and the goal of Ann is still to build an arbitrarily large word (Ben simply wants to limit the size of the word in this game). Grytczuk, Kozik, and Micek showed that Ann has a winning strategy for the nonrepetitive game if the alphabet is of size at least $6$ and for the erase-repetition game is the alphabet is of size at least $8$. In this article, we lower these bounds to respectively $4$ and $6$. The bound obtain by Grytczuk et al. relied on the so-called entropy compression and the previous bound by Pegden relied on some particular version of the Lov\'asz Local Lemma. We recently introduced a counting argument that can be applied to the same set of problems as entropy compression or the Lov\'asz Local Lemma and we use our method here. For these two games, we know that Ben has a winning strategy when the alphabet is of size at most 3, so our result for the nonrepetitive game is optimal, but we are not able to close the gap for the erase-repetition game.