{"paper":{"title":"Thresholds for Tic-Tac-Toe on Finite Affine Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"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.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alessandro Giannoni, Javier Lobillo-Olmedo, Luca Bastioni","submitted_at":"2026-05-06T21:24:33Z","abstract_excerpt":"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 $m$.\n  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 "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9058bd69bcbb87461ff7e51348eb305167e791fa81ddd843dce5e61c45af112a"},"source":{"id":"2605.05455","kind":"arxiv","version":3},"verdict":{"id":"7e33f614-abea-42d0-986b-0b502c785943","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T02:33:05.839915Z","strongest_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}.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.05455/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T09:41:11.904342Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.660608Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T13:33:55.281283Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"3687a38e5bb04ca6fccca4ee4215917fdd1ebed03b3e91129dd231d85cc51b41"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"4dbd2ccc499d032a5a8d62380a32a8331f54c5e0d738472f371c01b3bde8d17b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}