{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:7L4SAG4JH7B6L6CMHRU46MIUHN","short_pith_number":"pith:7L4SAG4J","schema_version":"1.0","canonical_sha256":"faf9201b893fc3e5f84c3c69cf31143b619509d480948cf7a454b7d8d5bc0af7","source":{"kind":"arxiv","id":"2406.10745","version":1},"attestation_state":"computed","paper":{"title":"Strong Brandt-Thomass\\'e Theorems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher, Joanna Polcyn, Tomasz {\\L}uczak","submitted_at":"2024-06-15T21:53:31Z","abstract_excerpt":"Solving a long standing conjecture of Erd\\H{o}s and Simonovits, Brandt and Thomass\\'e proved that the chromatic number of each triangle-free graph $G$ such that $\\delta(G)>|V(G)|/3$ is at most four. In fact, they showed the much stronger result that every maximal triangle-free graph $G$ satisfying this minimum degree condition is a blow-up of either an Andr\\'asfai or a Vega graph.\n  Here we establish the same structural conclusion on $G$ under the weaker assumption that for $m\\in\\{2, 3, 4\\}$ every sequence of $3m$ vertices has a subsequence of length $m+1$ with a common neighbour. In forthcomi"},"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":"2406.10745","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-06-15T21:53:31Z","cross_cats_sorted":[],"title_canon_sha256":"aa5dc49ead42011d4eeae11361649d31ac0a84424fdad62d445ad14654c02e93","abstract_canon_sha256":"4a826d46ccd35cc2f038b63ffafc754dc1aaf1bb31bcd51e86b1a352442cf205"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:32:21.436377Z","signature_b64":"lh5XNbK4/zarl7Ht8ovVj9Y3Flev+MY5sZxx8bTHqkwS3esT9aPbhlIc7RuucxFnup0ZWLicFL0rkptbRBDCCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"faf9201b893fc3e5f84c3c69cf31143b619509d480948cf7a454b7d8d5bc0af7","last_reissued_at":"2026-07-05T08:32:21.435860Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:32:21.435860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong Brandt-Thomass\\'e Theorems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher, Joanna Polcyn, Tomasz {\\L}uczak","submitted_at":"2024-06-15T21:53:31Z","abstract_excerpt":"Solving a long standing conjecture of Erd\\H{o}s and Simonovits, Brandt and Thomass\\'e proved that the chromatic number of each triangle-free graph $G$ such that $\\delta(G)>|V(G)|/3$ is at most four. In fact, they showed the much stronger result that every maximal triangle-free graph $G$ satisfying this minimum degree condition is a blow-up of either an Andr\\'asfai or a Vega graph.\n  Here we establish the same structural conclusion on $G$ under the weaker assumption that for $m\\in\\{2, 3, 4\\}$ every sequence of $3m$ vertices has a subsequence of length $m+1$ with a common neighbour. In forthcomi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.10745","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.10745/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":"2406.10745","created_at":"2026-07-05T08:32:21.435932+00:00"},{"alias_kind":"arxiv_version","alias_value":"2406.10745v1","created_at":"2026-07-05T08:32:21.435932+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.10745","created_at":"2026-07-05T08:32:21.435932+00:00"},{"alias_kind":"pith_short_12","alias_value":"7L4SAG4JH7B6","created_at":"2026-07-05T08:32:21.435932+00:00"},{"alias_kind":"pith_short_16","alias_value":"7L4SAG4JH7B6L6CM","created_at":"2026-07-05T08:32:21.435932+00:00"},{"alias_kind":"pith_short_8","alias_value":"7L4SAG4J","created_at":"2026-07-05T08:32:21.435932+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.01034","citing_title":"Ramsey with purple edges","ref_index":32,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN","json":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN.json","graph_json":"https://pith.science/api/pith-number/7L4SAG4JH7B6L6CMHRU46MIUHN/graph.json","events_json":"https://pith.science/api/pith-number/7L4SAG4JH7B6L6CMHRU46MIUHN/events.json","paper":"https://pith.science/paper/7L4SAG4J"},"agent_actions":{"view_html":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN","download_json":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN.json","view_paper":"https://pith.science/paper/7L4SAG4J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2406.10745&json=true","fetch_graph":"https://pith.science/api/pith-number/7L4SAG4JH7B6L6CMHRU46MIUHN/graph.json","fetch_events":"https://pith.science/api/pith-number/7L4SAG4JH7B6L6CMHRU46MIUHN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN/action/storage_attestation","attest_author":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN/action/author_attestation","sign_citation":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN/action/citation_signature","submit_replication":"https://pith.science/pith/7L4SAG4JH7B6L6CMHRU46MIUHN/action/replication_record"}},"created_at":"2026-07-05T08:32:21.435932+00:00","updated_at":"2026-07-05T08:32:21.435932+00:00"}