{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:OBVG7UMYRF7K6MWYGJDMCZVQYN","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":"5af10933f0b683a8802598670116d78bc872175a539987c5bcbf9e3bbcad1c88","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2009-11-09T17:55:06Z","title_canon_sha256":"3a707716e62f60512d7649824aa6032a2ffd59df0a54a227a52ff9cf8be87345"},"schema_version":"1.0","source":{"id":"0911.1741","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.1741","created_at":"2026-05-18T04:29:35Z"},{"alias_kind":"arxiv_version","alias_value":"0911.1741v4","created_at":"2026-05-18T04:29:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.1741","created_at":"2026-05-18T04:29:35Z"},{"alias_kind":"pith_short_12","alias_value":"OBVG7UMYRF7K","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"OBVG7UMYRF7K6MWY","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"OBVG7UMY","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:46033728e66f6fcf4645caaf8ab5abd5f7fe3a09d0b2b50aa8c916b80e033108","target":"graph","created_at":"2026-05-18T04:29:35Z","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":"Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a stable set meeting all maximum cliques. We strengthen this result, proving that such a stable set exists for any graph with omega > (2/3)(Delta+1). This is tight, i.e. the inequality in the statement must be strict. The proof relies on finding an independent transversal in a graph partitioned into vertex sets of unequal size.","authors_text":"Andrew D. King","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2009-11-09T17:55:06Z","title":"Hitting all maximum cliques with a stable set using lopsided independent transversals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.1741","kind":"arxiv","version":4},"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":"sha256:7c9bbcc77ca7a3d9290ee138dd56d725f8d6c444a3394c68e72473c6eb7da607","target":"record","created_at":"2026-05-18T04:29:35Z","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":"5af10933f0b683a8802598670116d78bc872175a539987c5bcbf9e3bbcad1c88","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2009-11-09T17:55:06Z","title_canon_sha256":"3a707716e62f60512d7649824aa6032a2ffd59df0a54a227a52ff9cf8be87345"},"schema_version":"1.0","source":{"id":"0911.1741","kind":"arxiv","version":4}},"canonical_sha256":"706a6fd198897eaf32d83246c166b0c3423181ac20d21a9d2eebe7312de9b0aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"706a6fd198897eaf32d83246c166b0c3423181ac20d21a9d2eebe7312de9b0aa","first_computed_at":"2026-05-18T04:29:35.104492Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:29:35.104492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fsod/LQS/zMsDEDMYwmGwAkvlkoy0gSrvwkZ6ytkeqk5PiUZxcVtbsdN1DtFyd4cVrTN3GhwA5xR1P8pe7/dAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:29:35.104949Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.1741","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c9bbcc77ca7a3d9290ee138dd56d725f8d6c444a3394c68e72473c6eb7da607","sha256:46033728e66f6fcf4645caaf8ab5abd5f7fe3a09d0b2b50aa8c916b80e033108"],"state_sha256":"e7708c5d9c78161ca5054a8ef7f1eeacb41149e4bff90b3efc928c3de7408054"}