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.

翻译：我们考虑两个玩家之间的两个游戏,两个玩家Ann和Ben通过在共享单词结尾处添加一个词来创建一个单词。在非重复式游戏中,Ben赢得了游戏,如果他能够创建一个长度的平方,至少4美元,而Ann赢得了游戏,如果她能够在此之前建立一个任意长的单词。在删除重复游戏中,当一个正方形出现时,广场的第二部分就会被抹去,而Ann的目标仍然是构建一个任意的大单词(Ben只是想要在这个游戏中限制单词的大小)。Grytczuk、Kozik和Micek显示,如果字母大小至少为6美元,Ann赢得了游戏的不重复游戏的胜利策略。在删除重复游戏中,Ann,如果她能够创建一个任意长一个长的单词,则Ann。在删除重复游戏中,如果她能够创建一个平方字的平方字面的游戏,至少8美元。在删除的游戏中,当Grytczuk et al(Ben) 想要一个任意的大单词(Beopy) 压缩和Pebilden(Pe) (Pe) (Be) (Pe) (Peal) (Be) (Be) (Be) (Be) (Bal) (Breal-legal) (B) (Bal-h) (Bal-h) (Bal-h) (Bal-lem) (Bal-h) (Lov) (我们知道tal-st) (n) (Lem-h) (Lemma) (n-h) (n-h) (n) (n) (n-h) (n) (n) (n) (n)) (Lem) (n) (n Lemm) (Lemma) (Lemma) (Lemma) (n) (n) (tal) (n) (n) (n) (n)) (n)) (n) (n) (n) (n) (n) (n) (n) (n) (tal) (n) (n) (tal) (n) (n) (n) (n) (n) (我们