{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:YKNFLVYQTVS6MMW2QUDQI7BZAC","short_pith_number":"pith:YKNFLVYQ","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"},"canonical_sha256":"c29a55d7109d65e632da8507047c39008fd6c594fd0f9de1a5c4b805eb530bbf","source":{"kind":"arxiv","id":"2207.10545","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2207.10545","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"arxiv_version","alias_value":"2207.10545v2","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.10545","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"pith_short_12","alias_value":"YKNFLVYQTVS6","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"pith_short_16","alias_value":"YKNFLVYQTVS6MMW2","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"pith_short_8","alias_value":"YKNFLVYQ","created_at":"2026-07-05T06:37:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:YKNFLVYQTVS6MMW2QUDQI7BZAC","target":"record","payload":{"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"},"canonical_sha256":"c29a55d7109d65e632da8507047c39008fd6c594fd0f9de1a5c4b805eb530bbf","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"},"source_kind":"arxiv","source_id":"2207.10545","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:37:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WKebcL9li3/H+KvGUH94zATqwyoan4YlJnZeOHyW0PtIutIcuGsCWS/Cyv2L4vLbFHY0989KuIbR4adhQ1bRCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T13:17:13.984860Z"},"content_sha256":"3121b157ce56c24b301861fd764ed8b8b2ac35c7e45605df7b2aaacedad9efe7","schema_version":"1.0","event_id":"sha256:3121b157ce56c24b301861fd764ed8b8b2ac35c7e45605df7b2aaacedad9efe7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:YKNFLVYQTVS6MMW2QUDQI7BZAC","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:37:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DNrqROROMQggcT+eQ8pOjZrMYm6j4GNjLh28CegwfAhzU7l4g+yZ51DilGnNWJO0y6dslezp11qSgC2AxGNYCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T13:17:13.985372Z"},"content_sha256":"6492aa529755641a04e4373fcece5c06fb8e6665fc588086225e5a04077470e6","schema_version":"1.0","event_id":"sha256:6492aa529755641a04e4373fcece5c06fb8e6665fc588086225e5a04077470e6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/bundle.json","state_url":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-04T13:17:13Z","links":{"resolver":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC","bundle":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/bundle.json","state":"https://pith.science/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YKNFLVYQTVS6MMW2QUDQI7BZAC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:YKNFLVYQTVS6MMW2QUDQI7BZAC","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":"6242792bc9efb21d50cb27f0bbfe31010f8131fad387971a4a6cf80a93ebb829","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-07-21T15:42:12Z","title_canon_sha256":"b0d517aeb5a7477e96664d92f4d4615ad18f8842960d4f6d08f64bba99abad97"},"schema_version":"1.0","source":{"id":"2207.10545","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2207.10545","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"arxiv_version","alias_value":"2207.10545v2","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.10545","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"pith_short_12","alias_value":"YKNFLVYQTVS6","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"pith_short_16","alias_value":"YKNFLVYQTVS6MMW2","created_at":"2026-07-05T06:37:10Z"},{"alias_kind":"pith_short_8","alias_value":"YKNFLVYQ","created_at":"2026-07-05T06:37:10Z"}],"graph_snapshots":[{"event_id":"sha256:6492aa529755641a04e4373fcece5c06fb8e6665fc588086225e5a04077470e6","target":"graph","created_at":"2026-07-05T06:37:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2207.10545/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Cassie Murley, Ce Chen, Grace McCourt, J\\'ozsef Balogh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-07-21T15:42:12Z","title":"Ramsey-Tur\\'an Problems with small independence numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.10545","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":"sha256:3121b157ce56c24b301861fd764ed8b8b2ac35c7e45605df7b2aaacedad9efe7","target":"record","created_at":"2026-07-05T06:37:10Z","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":"6242792bc9efb21d50cb27f0bbfe31010f8131fad387971a4a6cf80a93ebb829","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-07-21T15:42:12Z","title_canon_sha256":"b0d517aeb5a7477e96664d92f4d4615ad18f8842960d4f6d08f64bba99abad97"},"schema_version":"1.0","source":{"id":"2207.10545","kind":"arxiv","version":2}},"canonical_sha256":"c29a55d7109d65e632da8507047c39008fd6c594fd0f9de1a5c4b805eb530bbf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c29a55d7109d65e632da8507047c39008fd6c594fd0f9de1a5c4b805eb530bbf","first_computed_at":"2026-07-05T06:37:10.700843Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:37:10.700843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z5/9iUZ+0SozyKpbfdXpkJ7l2/yFtZ+iTv68fq1feRQij4lJJ3SVyvRt58SQlo8ZGlvLkRtwyPYd2XA5ax/RAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:37:10.701297Z","signed_message":"canonical_sha256_bytes"},"source_id":"2207.10545","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3121b157ce56c24b301861fd764ed8b8b2ac35c7e45605df7b2aaacedad9efe7","sha256:6492aa529755641a04e4373fcece5c06fb8e6665fc588086225e5a04077470e6"],"state_sha256":"20a5cf2c61b122b33f6a4e2a253d4b979f779f9b39059d43453dc854596a386a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZMKEkRmhF8fSG8LqK7wqnF5yy6KWIexTN3tt++2aHiMiiFJUw4qd8FN3sZftZzvHu3/kQyGB4CxbkR/zFNm1Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T13:17:13.989136Z","bundle_sha256":"18024c771eb21b5e13f6b1b64843fcbdd2ce2d76aaca44387f4fd79ed9df1e3f"}}