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Mubayi proved that \\ex_2(n,F)=(1/2+o(1))n and conjectured that \\ex_2(n, F)=\\floor{n/2} for sufficiently large n. Using a very sophisticated quasi-randomness argument, Keevash proved Mubayi's conjecture. Here we give a simple proof of Mubayi's conjecture by using a class of 3-graphs that we call rings. We also determine the Tur\\'an density of the family of rin"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1204.1927","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-09T17:06:15Z","cross_cats_sorted":[],"title_canon_sha256":"c935ab1c2e663ca1505b0fc683d5fda4ca65d8fc5e8a067f6aaf057fb364cf11","abstract_canon_sha256":"1a7728fd2b9f5a9861db61bef277b6a9c6c1c1eb887232b7efc16c65f7935b1e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:20:09.911853Z","signature_b64":"u2bCqnrVboPl2rOLcmFj8bWOKaT7fISxYI4OzjP+RgVGsRLILltS5No6j78Wo7cn7B83B6xbVZcsxnwWaykLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b169f57c1a823d4b4eafd2eeaa304eb64ad293bd1ef388990c11054a9479758d","last_reissued_at":"2026-05-18T03:20:09.911326Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:20:09.911326Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the co-degree threshold for the Fano plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Louis DeBiasio, Tao Jiang","submitted_at":"2012-04-09T17:06:15Z","abstract_excerpt":"Given a 3-graph H, let \\ex_2(n, H) denote the maximum value of the minimum codegree of a 3-graph on n vertices which does not contain a copy of H. 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