Thresholds for Tic-Tac-Toe on Finite Affine Spaces

We introduce an affine version of Tic-Tac-Toe played on the finite affine space $\mathbb{F}_q^m$. Two players alternately claim points, and the first player to occupy all points of an affine subspace of dimension $n$ wins. We call this the $(m,n)_q$-game. For fixed $n$ and $q$, we study how the outcome depends on the ambient dimension $m$. Using strategy stealing and a blocking-set interpretation, we show that every $(m,n)_q$-game is either a first-player win or a draw, and that the property of being a first-player win is monotone in $m$. This yields a threshold $T(n,q)$: the game is a draw for $m<T(n,q)$ and a first-player win for $m\ge T(n,q)$. We prove that this threshold is finite by applying the affine/vector-space Ramsey theorem of Graham, Leeb and Rothschild, and we obtain general lower bounds from the Erd\H{o}s-Selfridge criterion for Maker-Breaker games. In the binary case, we give a direct Fourier-analytic argument, combined with an inductive lifting method, which shows that $$T(n,2)\le 2^{n+1}.$$ We also determine several small cases, including $T(1,q)=2$ for $q\in\{2,3,4\}$ and $T(2,2)=4$, and we prove geometric lower bounds from explicit pairing strategies, such as $T(n,q)\ge n+2$ for every $n\ge 2$. Our results place affine Tic-Tac-Toe at the interface of strong positional games, finite geometry and Ramsey theory for finite affine spaces.