{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:D5FCGYKPJPOBQDV6WKMC5UK5L5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"22f584394135310cfa048d35a97c372d4b1f6616cfd340cf0800dc48258b62f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T18:01:15Z","title_canon_sha256":"b258adc8f8751570e92c62c65e72e76f43ccaf96bc997d0bd7783c3eb1d2d616"},"schema_version":"1.0","source":{"id":"1304.1036","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.1036","created_at":"2026-05-18T02:19:34Z"},{"alias_kind":"arxiv_version","alias_value":"1304.1036v2","created_at":"2026-05-18T02:19:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1036","created_at":"2026-05-18T02:19:34Z"},{"alias_kind":"pith_short_12","alias_value":"D5FCGYKPJPOB","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"D5FCGYKPJPOBQDV6","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"D5FCGYKP","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:31ae3c867ba9c6f52f18fd64336b5abbd5e4d0eac980738f8f1e3f698caae667","target":"graph","created_at":"2026-05-18T02:19:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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_","authors_text":"J\\'ozsef Balogh, Mikl\\'os Simonovits, Ping Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T18:01:15Z","title":"Phase transitions in the Ramsey-Tur\\'an theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1036","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8e7a9f818b658fd9fe0e72f22618dc956480a265483213ef60301495198b2d28","target":"record","created_at":"2026-05-18T02:19:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"22f584394135310cfa048d35a97c372d4b1f6616cfd340cf0800dc48258b62f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T18:01:15Z","title_canon_sha256":"b258adc8f8751570e92c62c65e72e76f43ccaf96bc997d0bd7783c3eb1d2d616"},"schema_version":"1.0","source":{"id":"1304.1036","kind":"arxiv","version":2}},"canonical_sha256":"1f4a23614f4bdc180ebeb2982ed15d5f51c13b96eef524ecb21e11206d987379","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1f4a23614f4bdc180ebeb2982ed15d5f51c13b96eef524ecb21e11206d987379","first_computed_at":"2026-05-18T02:19:34.362177Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:34.362177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EPhfCBDjursfaNBfBEIyXkQG9c8bujkSKQGsmurXp/Hu+Slwe89KPk81EBU/N8qwiswBgGEXayoRze37Z7XgDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:34.362683Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.1036","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8e7a9f818b658fd9fe0e72f22618dc956480a265483213ef60301495198b2d28","sha256:31ae3c867ba9c6f52f18fd64336b5abbd5e4d0eac980738f8f1e3f698caae667"],"state_sha256":"47cd90982157eddf589f5dc68b8313859bb728029be20a17e8ed0c88f98a8108"}