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For $H$ being a small clique, many results about $RT(n,H,f(n))$ are known and we focus our attention on $H=K_s$ for $s\\leq 13$. By applying Szemer\\'edi's Regularity Lemma, the dependent random choice method and some weighted Tur\\'an-type results, we prove that these cliques have the so-called phase transitions when $f(n)$ is around the inverse function of the off-diagonal Ramsey number of $K_r$ versus a large c"},"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":"2207.10545","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-07-21T15:42:12Z","cross_cats_sorted":[],"title_canon_sha256":"b0d517aeb5a7477e96664d92f4d4615ad18f8842960d4f6d08f64bba99abad97","abstract_canon_sha256":"6242792bc9efb21d50cb27f0bbfe31010f8131fad387971a4a6cf80a93ebb829"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:37:10.701297Z","signature_b64":"z5/9iUZ+0SozyKpbfdXpkJ7l2/yFtZ+iTv68fq1feRQij4lJJ3SVyvRt58SQlo8ZGlvLkRtwyPYd2XA5ax/RAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c29a55d7109d65e632da8507047c39008fd6c594fd0f9de1a5c4b805eb530bbf","last_reissued_at":"2026-07-05T06:37:10.700843Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:37:10.700843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ramsey-Tur\\'an Problems with small independence numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cassie Murley, Ce Chen, Grace McCourt, J\\'ozsef Balogh","submitted_at":"2022-07-21T15:42:12Z","abstract_excerpt":"Given a graph $H$ and a function $f(n)$, the Ramsey-Tur\\'an number $RT(n,H,f(n))$ is the maximum number of edges in an $n$-vertex $H$-free graph with independence number at most $f(n)$. For $H$ being a small clique, many results about $RT(n,H,f(n))$ are known and we focus our attention on $H=K_s$ for $s\\leq 13$. By applying Szemer\\'edi's Regularity Lemma, the dependent random choice method and some weighted Tur\\'an-type results, we prove that these cliques have the so-called phase transitions when $f(n)$ is around the inverse function of the off-diagonal Ramsey number of $K_r$ versus a large c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.10545","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2207.10545/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2207.10545","created_at":"2026-07-05T06:37:10.700892+00:00"},{"alias_kind":"arxiv_version","alias_value":"2207.10545v2","created_at":"2026-07-05T06:37:10.700892+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.10545","created_at":"2026-07-05T06:37:10.700892+00:00"},{"alias_kind":"pith_short_12","alias_value":"YKNFLVYQTVS6","created_at":"2026-07-05T06:37:10.700892+00:00"},{"alias_kind":"pith_short_16","alias_value":"YKNFLVYQTVS6MMW2","created_at":"2026-07-05T06:37:10.700892+00:00"},{"alias_kind":"pith_short_8","alias_value":"YKNFLVYQ","created_at":"2026-07-05T06:37:10.700892+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/YKNFLVYQTVS6MMW2QUDQI7BZAC","json":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC.json","graph_json":"https://pith.science/api/pith-number/YKNFLVYQTVS6MMW2QUDQI7BZAC/graph.json","events_json":"https://pith.science/api/pith-number/YKNFLVYQTVS6MMW2QUDQI7BZAC/events.json","paper":"https://pith.science/paper/YKNFLVYQ"},"agent_actions":{"view_html":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC","download_json":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC.json","view_paper":"https://pith.science/paper/YKNFLVYQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2207.10545&json=true","fetch_graph":"https://pith.science/api/pith-number/YKNFLVYQTVS6MMW2QUDQI7BZAC/graph.json","fetch_events":"https://pith.science/api/pith-number/YKNFLVYQTVS6MMW2QUDQI7BZAC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/action/storage_attestation","attest_author":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/action/author_attestation","sign_citation":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/action/citation_signature","submit_replication":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/action/replication_record"}},"created_at":"2026-07-05T06:37:10.700892+00:00","updated_at":"2026-07-05T06:37:10.700892+00:00"}