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We show that for all $n \\ge 6$, the maximum number of edges in an $F_{3,3}$-free 3-graph on $n$ vertices is $\\binom{n}{3} - \\binom{\\lfloor n/2 \\rfloor}{3} - \\binom{\\lceil n/2 \\rceil}{3}$. This sharpens results of Zhou and of the second author and R\\\"odl."},"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":"1102.2141","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-10T15:15:16Z","cross_cats_sorted":[],"title_canon_sha256":"a21c2370e66a0f94afc6d4cce834552af3c741c16c5eead158a9874f8a493c54","abstract_canon_sha256":"2583603ad7743db31292d4cd92282101885e1d97049e6c39df3d80d3e09ee440"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:29:34.167646Z","signature_b64":"fZ3+enJQZrtZe9z5e8O0bWDTiBXRVQUNQbOUNb5sDmZGxJF8mkD1sNKjJpA8ZC6p0UZv6VRwwskZIFnoPzPqCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b637da97b3a34a346436b88f8ec5a9f64026c8200d060d492279063d49259d83","last_reissued_at":"2026-05-18T04:29:34.166986Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:29:34.166986Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Tur\\'an number of $F_{3,3}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Peter Keevash","submitted_at":"2011-02-10T15:15:16Z","abstract_excerpt":"Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. 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