{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:D5FCGYKPJPOBQDV6WKMC5UK5L5","short_pith_number":"pith:D5FCGYKP","schema_version":"1.0","canonical_sha256":"1f4a23614f4bdc180ebeb2982ed15d5f51c13b96eef524ecb21e11206d987379","source":{"kind":"arxiv","id":"1304.1036","version":2},"attestation_state":"computed","paper":{"title":"Phase transitions in the Ramsey-Tur\\'an theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'ozsef Balogh, Mikl\\'os Simonovits, Ping Hu","submitted_at":"2013-04-03T18:01:15Z","abstract_excerpt":"Let $f(n)$ be a function and $L$ be a graph. Denote by $RT(n,L,f(n))$ the maximum number of edges of an $L$-free graph on $n$ vertices with independence number less than $f(n)$. Erd\\H os and S\\'os asked if $RT\\left(n, K_5, c\\sqrt{n}\\right) = o(n^2)$ for some constant $c$. We answer this question by proving the stronger $RT\\left(n, K_5, o\\left(\\sqrt{n\\log n}\\right)\\right) = o(n^2)$. It is known that $RT \\left(n, K_5, c \\sqrt{n\\log n} \\right) = n^2/4+o(n^2)$ for $c>1$, so one can say that $K_5$ has a Ramsey-Tur\\'an phase transition at $c\\sqrt{n\\log n}$. We extend this result to several other $K_"},"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":"1304.1036","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T18:01:15Z","cross_cats_sorted":[],"title_canon_sha256":"b258adc8f8751570e92c62c65e72e76f43ccaf96bc997d0bd7783c3eb1d2d616","abstract_canon_sha256":"22f584394135310cfa048d35a97c372d4b1f6616cfd340cf0800dc48258b62f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:34.362683Z","signature_b64":"EPhfCBDjursfaNBfBEIyXkQG9c8bujkSKQGsmurXp/Hu+Slwe89KPk81EBU/N8qwiswBgGEXayoRze37Z7XgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f4a23614f4bdc180ebeb2982ed15d5f51c13b96eef524ecb21e11206d987379","last_reissued_at":"2026-05-18T02:19:34.362177Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:34.362177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Phase transitions in the Ramsey-Tur\\'an theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'ozsef Balogh, Mikl\\'os Simonovits, Ping Hu","submitted_at":"2013-04-03T18:01:15Z","abstract_excerpt":"Let $f(n)$ be a function and $L$ be a graph. Denote by $RT(n,L,f(n))$ the maximum number of edges of an $L$-free graph on $n$ vertices with independence number less than $f(n)$. Erd\\H os and S\\'os asked if $RT\\left(n, K_5, c\\sqrt{n}\\right) = o(n^2)$ for some constant $c$. We answer this question by proving the stronger $RT\\left(n, K_5, o\\left(\\sqrt{n\\log n}\\right)\\right) = o(n^2)$. It is known that $RT \\left(n, K_5, c \\sqrt{n\\log n} \\right) = n^2/4+o(n^2)$ for $c>1$, so one can say that $K_5$ has a Ramsey-Tur\\'an phase transition at $c\\sqrt{n\\log n}$. We extend this result to several other $K_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1036","kind":"arxiv","version":2},"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":"1304.1036","created_at":"2026-05-18T02:19:34.362279+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.1036v2","created_at":"2026-05-18T02:19:34.362279+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1036","created_at":"2026-05-18T02:19:34.362279+00:00"},{"alias_kind":"pith_short_12","alias_value":"D5FCGYKPJPOB","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_16","alias_value":"D5FCGYKPJPOBQDV6","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_8","alias_value":"D5FCGYKP","created_at":"2026-05-18T12:27:40.988391+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/D5FCGYKPJPOBQDV6WKMC5UK5L5","json":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5.json","graph_json":"https://pith.science/api/pith-number/D5FCGYKPJPOBQDV6WKMC5UK5L5/graph.json","events_json":"https://pith.science/api/pith-number/D5FCGYKPJPOBQDV6WKMC5UK5L5/events.json","paper":"https://pith.science/paper/D5FCGYKP"},"agent_actions":{"view_html":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5","download_json":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5.json","view_paper":"https://pith.science/paper/D5FCGYKP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.1036&json=true","fetch_graph":"https://pith.science/api/pith-number/D5FCGYKPJPOBQDV6WKMC5UK5L5/graph.json","fetch_events":"https://pith.science/api/pith-number/D5FCGYKPJPOBQDV6WKMC5UK5L5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5/action/storage_attestation","attest_author":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5/action/author_attestation","sign_citation":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5/action/citation_signature","submit_replication":"https://pith.science/pith/D5FCGYKPJPOBQDV6WKMC5UK5L5/action/replication_record"}},"created_at":"2026-05-18T02:19:34.362279+00:00","updated_at":"2026-05-18T02:19:34.362279+00:00"}