{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ZWKQH4MW2FATCSDTSJZ34S6JBO","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":"5bd16c491f823b52b80dc8153aba82ab1c74b6645d1a590786b2b5ce1b5e1780","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-12T02:18:21Z","title_canon_sha256":"8e1e0a4c65b44e9f87fff03ee3aedc72d0369a2cd25aeeedfb4a478cf6b8f983"},"schema_version":"1.0","source":{"id":"1101.2257","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.2257","created_at":"2026-05-18T04:31:43Z"},{"alias_kind":"arxiv_version","alias_value":"1101.2257v1","created_at":"2026-05-18T04:31:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.2257","created_at":"2026-05-18T04:31:43Z"},{"alias_kind":"pith_short_12","alias_value":"ZWKQH4MW2FAT","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"ZWKQH4MW2FATCSDT","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"ZWKQH4MW","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:e5592703c09a2bcf1f31ee8ea544c70a6c9ac742c1ce7a04ac152b10a06654e3","target":"graph","created_at":"2026-05-18T04:31:43Z","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":"Let $G(X,Y)$ be a connected, non-complete bipartite graph with $|X|\\leq |Y|$. An independent set $A$ of $G(X,Y)$ is said to be trivial if $A\\subseteq X$ or $A\\subseteq Y$. Otherwise, $A$ is nontrivial. By $\\alpha(X,Y)$ we denote the size of maximal-sized nontrivial independent sets of $G(X,Y)$. We prove that if the automorphism group of $G(X,Y)$ is transitive on $X$ and $Y$, then $\\alpha(X,Y)=|Y|-d(X)+1$, where $d(X)$ is the common degree of vertices in $X$. We also give the structures of maximal-sized nontrivial independent sets of $G(X,Y)$. As applications of this result, we give the upper b","authors_text":"Huajun Zhang, Jun Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-12T02:18:21Z","title":"Nontrivial independent sets of bipartite graphs and cross-intersecting families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2257","kind":"arxiv","version":1},"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":"sha256:80a84570f05286b620397411be80088f7333fad48e05f76b0de4ac6f36783454","target":"record","created_at":"2026-05-18T04:31:43Z","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":"5bd16c491f823b52b80dc8153aba82ab1c74b6645d1a590786b2b5ce1b5e1780","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-12T02:18:21Z","title_canon_sha256":"8e1e0a4c65b44e9f87fff03ee3aedc72d0369a2cd25aeeedfb4a478cf6b8f983"},"schema_version":"1.0","source":{"id":"1101.2257","kind":"arxiv","version":1}},"canonical_sha256":"cd9503f196d1413148739273be4bc90b8084c95da2e750fea7b6c576ae560983","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd9503f196d1413148739273be4bc90b8084c95da2e750fea7b6c576ae560983","first_computed_at":"2026-05-18T04:31:43.256924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:43.256924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AOMm1sjqz2I721QrXXSK468a7Qaiwp8uiQ0V/getKePmFM7hmdpJnaJMa9z9T7eq4U5qcdCpF+C407NVT0t5Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:43.257386Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.2257","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80a84570f05286b620397411be80088f7333fad48e05f76b0de4ac6f36783454","sha256:e5592703c09a2bcf1f31ee8ea544c70a6c9ac742c1ce7a04ac152b10a06654e3"],"state_sha256":"9af8587f54d13d71b209d6485361ac62dee9abc86a670f369d76b8864693cd05"}