Abstract: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 Erdős-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.
| Subjects: | Combinatorics (math.CO) |
| MSC classes: | 91A46, 05B25, 05D10, 51E20 |
| Cite as: | arXiv:2605.05455 [math.CO] |
| (or arXiv:2605.05455v3 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.05455 arXiv-issued DOI via DataCite |
From: Alessandro Giannoni [view email]
[v1]
Wed, 6 May 2026 21:24:33 UTC (22 KB)
[v2]
Fri, 8 May 2026 18:47:09 UTC (24 KB)
[v3]
Fri, 22 May 2026 14:35:49 UTC (25 KB)
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