{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:6NYCZV4O52L74NDGD6R7JIHY3Q","short_pith_number":"pith:6NYCZV4O","schema_version":"1.0","canonical_sha256":"f3702cd78eee97fe34661fa3f4a0f8dc31bfd3583e1501411266f33221dc90ce","source":{"kind":"arxiv","id":"1309.0038","version":2},"attestation_state":"computed","paper":{"title":"The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Goedgebeur, Stanis{\\l}aw P. Radziszowski","submitted_at":"2013-08-30T22:09:35Z","abstract_excerpt":"Using computer algorithms we establish that the Ramsey number $R(3,K_{10}-e)$ is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of $R(3,K_k-e)$ for $11 \\le k \\le 16$, and show by construction a new lower bound $55 \\le R(3,K_{13}-e)$.\n  The new upper bounds on $R(3,K_k-e)$ are obtained by using the values and lower bounds on $e(3,K_l-e,n)$ for $l \\le k$, where $e(3,K_k-e,n)$ is the minimum number of edges in any triangle-free graph on $n$ vertices without $K_k-e$ in the complement. We complete the computation of th"},"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":"1309.0038","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-30T22:09:35Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"b3df4469826ece163f58695c99b33017f01167b4b334bfc8e3cd3ceeb1151116","abstract_canon_sha256":"f51e22dca5bc83107e7209ded69b1a3a7649ddb7041be6749fa695c917b3632a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:54.498488Z","signature_b64":"dEVNNHU4+xqjOlplb1Xes17cEIIliPSPqABQttywD7cJ98DeqCeMHGAQUFtQ0tK9aDJBkI+LeAZXphRE3JrZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3702cd78eee97fe34661fa3f4a0f8dc31bfd3583e1501411266f33221dc90ce","last_reissued_at":"2026-05-18T03:06:54.497696Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:54.497696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Goedgebeur, Stanis{\\l}aw P. Radziszowski","submitted_at":"2013-08-30T22:09:35Z","abstract_excerpt":"Using computer algorithms we establish that the Ramsey number $R(3,K_{10}-e)$ is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of $R(3,K_k-e)$ for $11 \\le k \\le 16$, and show by construction a new lower bound $55 \\le R(3,K_{13}-e)$.\n  The new upper bounds on $R(3,K_k-e)$ are obtained by using the values and lower bounds on $e(3,K_l-e,n)$ for $l \\le k$, where $e(3,K_k-e,n)$ is the minimum number of edges in any triangle-free graph on $n$ vertices without $K_k-e$ in the complement. We complete the computation of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0038","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":"1309.0038","created_at":"2026-05-18T03:06:54.497830+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.0038v2","created_at":"2026-05-18T03:06:54.497830+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0038","created_at":"2026-05-18T03:06:54.497830+00:00"},{"alias_kind":"pith_short_12","alias_value":"6NYCZV4O52L7","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"6NYCZV4O52L74NDG","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"6NYCZV4O","created_at":"2026-05-18T12:27:36.564083+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/6NYCZV4O52L74NDGD6R7JIHY3Q","json":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q.json","graph_json":"https://pith.science/api/pith-number/6NYCZV4O52L74NDGD6R7JIHY3Q/graph.json","events_json":"https://pith.science/api/pith-number/6NYCZV4O52L74NDGD6R7JIHY3Q/events.json","paper":"https://pith.science/paper/6NYCZV4O"},"agent_actions":{"view_html":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q","download_json":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q.json","view_paper":"https://pith.science/paper/6NYCZV4O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.0038&json=true","fetch_graph":"https://pith.science/api/pith-number/6NYCZV4O52L74NDGD6R7JIHY3Q/graph.json","fetch_events":"https://pith.science/api/pith-number/6NYCZV4O52L74NDGD6R7JIHY3Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q/action/storage_attestation","attest_author":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q/action/author_attestation","sign_citation":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q/action/citation_signature","submit_replication":"https://pith.science/pith/6NYCZV4O52L74NDGD6R7JIHY3Q/action/replication_record"}},"created_at":"2026-05-18T03:06:54.497830+00:00","updated_at":"2026-05-18T03:06:54.497830+00:00"}