{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6NYCZV4O52L74NDGD6R7JIHY3Q","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":"f51e22dca5bc83107e7209ded69b1a3a7649ddb7041be6749fa695c917b3632a","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-30T22:09:35Z","title_canon_sha256":"b3df4469826ece163f58695c99b33017f01167b4b334bfc8e3cd3ceeb1151116"},"schema_version":"1.0","source":{"id":"1309.0038","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0038","created_at":"2026-05-18T03:06:54Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0038v2","created_at":"2026-05-18T03:06:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0038","created_at":"2026-05-18T03:06:54Z"},{"alias_kind":"pith_short_12","alias_value":"6NYCZV4O52L7","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6NYCZV4O52L74NDG","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6NYCZV4O","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:79327c2c4f8d3ef3958fda4c6dd88343799c1c7c983e962767f579edb19f50e4","target":"graph","created_at":"2026-05-18T03:06:54Z","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":"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","authors_text":"Jan Goedgebeur, Stanis{\\l}aw P. Radziszowski","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-30T22:09:35Z","title":"The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0038","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":"sha256:198776c2e15971454c57e809021fb15dd249d2bfa59aedb33447f9f974de80ab","target":"record","created_at":"2026-05-18T03:06:54Z","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":"f51e22dca5bc83107e7209ded69b1a3a7649ddb7041be6749fa695c917b3632a","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-30T22:09:35Z","title_canon_sha256":"b3df4469826ece163f58695c99b33017f01167b4b334bfc8e3cd3ceeb1151116"},"schema_version":"1.0","source":{"id":"1309.0038","kind":"arxiv","version":2}},"canonical_sha256":"f3702cd78eee97fe34661fa3f4a0f8dc31bfd3583e1501411266f33221dc90ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3702cd78eee97fe34661fa3f4a0f8dc31bfd3583e1501411266f33221dc90ce","first_computed_at":"2026-05-18T03:06:54.497696Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:06:54.497696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dEVNNHU4+xqjOlplb1Xes17cEIIliPSPqABQttywD7cJ98DeqCeMHGAQUFtQ0tK9aDJBkI+LeAZXphRE3JrZCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:06:54.498488Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.0038","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:198776c2e15971454c57e809021fb15dd249d2bfa59aedb33447f9f974de80ab","sha256:79327c2c4f8d3ef3958fda4c6dd88343799c1c7c983e962767f579edb19f50e4"],"state_sha256":"39bd2da442bf7608014b54d55656bd8384c8c5b98a6a05baea49bf613cb22e9b"}