Pith Number

pith:FCS5YOBV

pith:2026:FCS5YOBVIZDJ5AWLQEA23Y53MJ

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Thresholds for Tic-Tac-Toe on Finite Affine Spaces

Alessandro Giannoni, Javier Lobillo-Olmedo, Luca Bastioni

For fixed n and q, affine Tic-Tac-Toe on F_q^m has a finite threshold T(n,q) above which the first player always wins.

arxiv:2605.05455 v3 · 2026-05-06 · math.CO

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4 Citations open

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Claims

C1strongest claim

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≥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)≤2^{n+1}.

C2weakest assumption

The monotonicity of first-player wins in m and the direct applicability of the Graham-Leeb-Rothschild affine Ramsey theorem to guarantee a finite threshold without needing explicit Ramsey numbers for the game-specific coloring.

C3one line summary

Affine Tic-Tac-Toe on F_q^m has a finite threshold T(n,q) separating draws from first-player wins, with T(n,2) at most 2^{n+1} and exact values for several small cases.

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-25T02:02:16.351541Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

28a5dc383546469e82cb8101ade3bb62720cc3b9e6c63cc2490d3bb0cfaae4c6

Aliases

arxiv: 2605.05455 · arxiv_version: 2605.05455v3 · doi: 10.48550/arxiv.2605.05455 · pith_short_12: FCS5YOBVIZDJ · pith_short_16: FCS5YOBVIZDJ5AWL · pith_short_8: FCS5YOBV

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/FCS5YOBVIZDJ5AWLQEA23Y53MJ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 28a5dc383546469e82cb8101ade3bb62720cc3b9e6c63cc2490d3bb0cfaae4c6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "1e939c4317cf7c456ad401797366869d2d06593b3951efde6dfab7483df2822d",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-06T21:24:33Z",
    "title_canon_sha256": "e3a017adfdb9808ccffdd3e2cb40733140f443c68f089db29defc80050b4828c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.05455",
    "kind": "arxiv",
    "version": 3
  }
}