Almost optimal pairing strategy for Tic-Tac-Toe with numerous directions

Abstract

We show that there is an $m=2n+o(n)$, such that, in the Maker-Breaker game played on ${\mathbb{Z}}^d$ where Maker needs to put at least $m$ of his marks consecutively in one of $n$ given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg who showed that such a pairing strategy exits if $m\ge 3n$. A simple argument shows that $m$ has to be at least $2n+1$ if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.