{"paper":{"title":"Random Tur\\'an and counting results for general position sets over finite fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiaxi Nie, Ji Zeng, Xizhi Liu, Yaobin Chen","submitted_at":"2023-09-14T14:30:47Z","abstract_excerpt":"Let $\\alpha(\\mathbb{F}_q^d,p)$ denote the maximum size of a general position set in a $p$-random subset of $\\mathbb{F}_q^d$. We determine the order of magnitude of $\\alpha(\\mathbb{F}_q^2,p)$ up to polylogarithmic factors for all possible values of $p$, improving the previous results obtained by Roche-Newton--Warren and Bhowmick--Roche-Newton. For $d \\ge 3$ we prove upper bounds for $\\alpha(\\mathbb{F}_q^d,p)$ that are essentially tight within certain ranges for $p$.\n  We establish the upper bound $2^{(1+o(1))q}$ for the number of general position sets in $\\mathbb{F}_q^d$, which matches the triv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.07744","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2309.07744/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}