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With $G_{n,p}$ the usual (\"binomial\" or \"Erd\\H{o}s-R\\'enyi\") random graph, we show:\n  For each fixed r there is a C such that if \\[ p=p(n) > Cn^{-\\tfrac{2}{r+1}}\\log^{\\tfrac{2}{(r+1)(r-2)}}n, \\] then $\\Pr(t_r(G_{n,p})=b_r(G_{n,p}))\\rightarrow 1$ as $n\\rightarrow\\infty$.\n  This is best possible (apart from the value of $C$) and settles a question first conside"},"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":"1501.01340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-07T00:46:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2691a375b3375c49b6c5b2e64fff36fa0ca6fb1b13cc81c85543233ab6fa5c15","abstract_canon_sha256":"7efd0708d95e23a67871117315b43cd34c878e109156cfb4ca61b000083edce6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:54.474584Z","signature_b64":"cvfaTE/SfwpRben99+i0v2iGNBFVr6XANVvf7xN/rUl0GLD0qHwVPjppVdbO0ro40f07BGyH1xi4C6fQVnc+DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6aec373edccb7fd19163017f0c4cf73c48b6a7e149ef605f83d550d284734c7","last_reissued_at":"2026-05-18T02:29:54.474073Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:54.474073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tur\\'an's Theorem for random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Bobby DeMarco, Jeff Kahn","submitted_at":"2015-01-07T00:46:17Z","abstract_excerpt":"For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \\geq b_r(G)$ for any $G$, and Tur\\'an's Theorem says that equality holds for complete graphs. With $G_{n,p}$ the usual (\"binomial\" or \"Erd\\H{o}s-R\\'enyi\") random graph, we show:\n  For each fixed r there is a C such that if \\[ p=p(n) > Cn^{-\\tfrac{2}{r+1}}\\log^{\\tfrac{2}{(r+1)(r-2)}}n, \\] then $\\Pr(t_r(G_{n,p})=b_r(G_{n,p}))\\rightarrow 1$ as $n\\rightarrow\\infty$.\n  This is best possible (apart from the value of $C$) and settles a question first conside"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01340","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1501.01340","created_at":"2026-05-18T02:29:54.474145+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.01340v1","created_at":"2026-05-18T02:29:54.474145+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01340","created_at":"2026-05-18T02:29:54.474145+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y2XMG47NZS37","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y2XMG47NZS372GIW","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y2XMG47N","created_at":"2026-05-18T12:29:50.041715+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP","json":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP.json","graph_json":"https://pith.science/api/pith-number/Y2XMG47NZS372GIWGAL7BRGPOP/graph.json","events_json":"https://pith.science/api/pith-number/Y2XMG47NZS372GIWGAL7BRGPOP/events.json","paper":"https://pith.science/paper/Y2XMG47N"},"agent_actions":{"view_html":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP","download_json":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP.json","view_paper":"https://pith.science/paper/Y2XMG47N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.01340&json=true","fetch_graph":"https://pith.science/api/pith-number/Y2XMG47NZS372GIWGAL7BRGPOP/graph.json","fetch_events":"https://pith.science/api/pith-number/Y2XMG47NZS372GIWGAL7BRGPOP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP/action/storage_attestation","attest_author":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP/action/author_attestation","sign_citation":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP/action/citation_signature","submit_replication":"https://pith.science/pith/Y2XMG47NZS372GIWGAL7BRGPOP/action/replication_record"}},"created_at":"2026-05-18T02:29:54.474145+00:00","updated_at":"2026-05-18T02:29:54.474145+00:00"}