{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JBETWCJZNNLYYULGZWMB4FCVAC","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":"69fc467ff22337eccc74bf3603480a2c2ad0aacd87f414b188ccf27e47f12a74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-12T22:19:43Z","title_canon_sha256":"527533f70b7e2c0c7bbc1014b9b32feba7051a3f80338e7cf408ed73b71610a7"},"schema_version":"1.0","source":{"id":"1203.2667","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.2667","created_at":"2026-05-18T02:28:32Z"},{"alias_kind":"arxiv_version","alias_value":"1203.2667v2","created_at":"2026-05-18T02:28:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.2667","created_at":"2026-05-18T02:28:32Z"},{"alias_kind":"pith_short_12","alias_value":"JBETWCJZNNLY","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JBETWCJZNNLYYULG","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JBETWCJZ","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:bb2810e3749b914ea76fe574d9d3f113445015323c8f66625148adf361e2d5d5","target":"graph","created_at":"2026-05-18T02:28:32Z","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 $G$ be a $k$-partite graph with $n$ vertices in parts such that each vertex is adjacent to at least $\\delta^*(G)$ vertices in each of the other parts. Magyar and Martin \\cite{MaMa} proved that for $k=3$, if $\\delta^*(G)\\ge 2/3n $ and $n$ is sufficiently large, then $G$ contains a $K_3$-factor (a spanning subgraph consisting of $n$ vertex-disjoint copies of $K_3$) except that $G$ is one particular graph. Martin and Szemer\\'edi \\cite{MaSz} proved that $G$ contains a $K_4$-factor when $\\delta^*(G)\\ge 3/4n$ and $n$ is sufficiently large. Both results were proved by the Regularity Lemma. In thi","authors_text":"Jie Han, Yi Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-12T22:19:43Z","title":"On multipartite Hajnal-Szemer\\'edi theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.2667","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:dada6bf30780c89a18c6cef5c36a43fa9d1398a5d1ca9f68164a74ff8aa98b54","target":"record","created_at":"2026-05-18T02:28:32Z","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":"69fc467ff22337eccc74bf3603480a2c2ad0aacd87f414b188ccf27e47f12a74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-12T22:19:43Z","title_canon_sha256":"527533f70b7e2c0c7bbc1014b9b32feba7051a3f80338e7cf408ed73b71610a7"},"schema_version":"1.0","source":{"id":"1203.2667","kind":"arxiv","version":2}},"canonical_sha256":"48493b09396b578c5166cd981e145500b2a04c53810f95d00cbefe3260107227","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48493b09396b578c5166cd981e145500b2a04c53810f95d00cbefe3260107227","first_computed_at":"2026-05-18T02:28:32.144965Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:32.144965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cwhROfjCKch1yucgYaRJrfNBayCUBM265Xaz+y1wBfeEiiz37vkL5UBdLMaNH9nXt4Rujxv9Mix5SVZWod7yBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:32.145653Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.2667","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dada6bf30780c89a18c6cef5c36a43fa9d1398a5d1ca9f68164a74ff8aa98b54","sha256:bb2810e3749b914ea76fe574d9d3f113445015323c8f66625148adf361e2d5d5"],"state_sha256":"c654e0b9f5ce72e7cce9f77eb5a85c5e4eec03c945bd3b94d7992d956fc57fff"}