{"paper":{"title":"Finite Hypergraph Families with Rich Extremal Tur\\'an Constructions via Mixing Patterns","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oleg Pikhurko, Xizhi Liu","submitted_at":"2022-12-16T18:25:49Z","abstract_excerpt":"We prove that, for any finite set of minimal $r$-graph patterns, there is a finite family $\\mathcal F$ of forbidden $r$-graphs such that the extremal Tur\\'an constructions for $\\mathcal F$ are precisely the maximum $r$-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author \\cite{PI14}, where the above statement was established for a single pattern.\n  We present two applications of this result. First, we construct a finite family $\\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\\mathcal F$-fre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.08636","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/2212.08636/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"}