{"paper":{"title":"The no-three-in-line problem on a torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Andrew Groot, Bart Snapp, Deven Pandya, Jim Fowler","submitted_at":"2012-03-29T17:41:02Z","abstract_excerpt":"Let $T(\\Z_m \\times \\Z_n)$ denote the maximal number of points that can be placed on an $m \\times n$ discrete torus with \"no three in a line,\" meaning no three in a coset of a cyclic subgroup of $\\Z_m \\times \\Z_n$. By proving upper bounds and providing explicit constructions, for distinct primes $p$ and $q$, we show that $T(\\Z_p \\times \\Z_{p^2}) = 2p$ and $T(\\Z_p \\times \\Z_{pq}) = p+1$. Via Gr\\\"obner bases, we compute $T(\\Z_m \\times \\Z_n)$ for $2 \\leq m \\leq 7$ and $2 \\leq n \\leq 19$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6604","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}