{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:EKY5ANCAP6FU5N2MPET5KFTVV4","short_pith_number":"pith:EKY5ANCA","canonical_record":{"source":{"id":"1307.4803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-17T22:46:10Z","cross_cats_sorted":[],"title_canon_sha256":"9ba34b5ceb278eb7fd497beb567aec8b7062a0ccfa88180233bad906e6734f9d","abstract_canon_sha256":"dfea46986b442657862b5b1225f87c3489986dec5a97538ca9750ec1b8a0666d"},"schema_version":"1.0"},"canonical_sha256":"22b1d034407f8b4eb74c7927d51675af07fe0a50fca2b7fbb20d843c90be98b4","source":{"kind":"arxiv","id":"1307.4803","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.4803","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"arxiv_version","alias_value":"1307.4803v1","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.4803","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"pith_short_12","alias_value":"EKY5ANCAP6FU","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EKY5ANCAP6FU5N2M","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EKY5ANCA","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:EKY5ANCAP6FU5N2MPET5KFTVV4","target":"record","payload":{"canonical_record":{"source":{"id":"1307.4803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-17T22:46:10Z","cross_cats_sorted":[],"title_canon_sha256":"9ba34b5ceb278eb7fd497beb567aec8b7062a0ccfa88180233bad906e6734f9d","abstract_canon_sha256":"dfea46986b442657862b5b1225f87c3489986dec5a97538ca9750ec1b8a0666d"},"schema_version":"1.0"},"canonical_sha256":"22b1d034407f8b4eb74c7927d51675af07fe0a50fca2b7fbb20d843c90be98b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:08.634652Z","signature_b64":"jF/wxFVeX3ZtsDTzL7G5GcmC1TneTlztnOtV8QmdLVoBr5gBOKYG1GzCBctplOo4/F7azxpbz2jwnWpdX8ZiBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22b1d034407f8b4eb74c7927d51675af07fe0a50fca2b7fbb20d843c90be98b4","last_reissued_at":"2026-05-18T03:18:08.633988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:08.633988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.4803","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RDiEcfiAi0IrVBil82JbnDsp9OrOj/IrkUAmvKze0Lx++MOgiGd06YEZCjhxl/HYNBweCZhBh9Gk+701Kp2EDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:48:11.397073Z"},"content_sha256":"c6ff336d20cb6a41f711f4737dfeed5a9c078f3ed39009d738e496b87e118502","schema_version":"1.0","event_id":"sha256:c6ff336d20cb6a41f711f4737dfeed5a9c078f3ed39009d738e496b87e118502"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:EKY5ANCAP6FU5N2MPET5KFTVV4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An extension of the Hajnal-Szemeredi theorem to directed graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej Czygrinow, H.A. Kierstead, Louis DeBiasio, Theodore Molla","submitted_at":"2013-07-17T22:46:10Z","abstract_excerpt":"Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph D with |D|=ks and minimum (total) degree at least 2k(s-1)-1 contains k vertex disjoint transitive tournaments on s vertices. Our result implies the Hajnal-Szemeredi Theorem, and the degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.4803","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rR9S3hocjfUAEgcg02CsOybc6X3HM1UTFFsMzyvYcx/Sqs7oCyuWc+KlqCXqldfKzuUUjGNmfAQ7NFJnRsJrAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:48:11.397428Z"},"content_sha256":"1995e487af6a83797f9acf81e42ab227cddde5ae13ea61a40086a9a5a1392b1f","schema_version":"1.0","event_id":"sha256:1995e487af6a83797f9acf81e42ab227cddde5ae13ea61a40086a9a5a1392b1f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EKY5ANCAP6FU5N2MPET5KFTVV4/bundle.json","state_url":"https://pith.science/pith/EKY5ANCAP6FU5N2MPET5KFTVV4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EKY5ANCAP6FU5N2MPET5KFTVV4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T21:48:11Z","links":{"resolver":"https://pith.science/pith/EKY5ANCAP6FU5N2MPET5KFTVV4","bundle":"https://pith.science/pith/EKY5ANCAP6FU5N2MPET5KFTVV4/bundle.json","state":"https://pith.science/pith/EKY5ANCAP6FU5N2MPET5KFTVV4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EKY5ANCAP6FU5N2MPET5KFTVV4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EKY5ANCAP6FU5N2MPET5KFTVV4","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":"dfea46986b442657862b5b1225f87c3489986dec5a97538ca9750ec1b8a0666d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-17T22:46:10Z","title_canon_sha256":"9ba34b5ceb278eb7fd497beb567aec8b7062a0ccfa88180233bad906e6734f9d"},"schema_version":"1.0","source":{"id":"1307.4803","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.4803","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"arxiv_version","alias_value":"1307.4803v1","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.4803","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"pith_short_12","alias_value":"EKY5ANCAP6FU","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EKY5ANCAP6FU5N2M","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EKY5ANCA","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:1995e487af6a83797f9acf81e42ab227cddde5ae13ea61a40086a9a5a1392b1f","target":"graph","created_at":"2026-05-18T03:18:08Z","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":"Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph D with |D|=ks and minimum (total) degree at least 2k(s-1)-1 contains k vertex disjoint transitive tournaments on s vertices. Our result implies the Hajnal-Szemeredi Theorem, and the degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjec","authors_text":"Andrzej Czygrinow, H.A. Kierstead, Louis DeBiasio, Theodore Molla","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-17T22:46:10Z","title":"An extension of the Hajnal-Szemeredi theorem to directed graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.4803","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:c6ff336d20cb6a41f711f4737dfeed5a9c078f3ed39009d738e496b87e118502","target":"record","created_at":"2026-05-18T03:18:08Z","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":"dfea46986b442657862b5b1225f87c3489986dec5a97538ca9750ec1b8a0666d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-17T22:46:10Z","title_canon_sha256":"9ba34b5ceb278eb7fd497beb567aec8b7062a0ccfa88180233bad906e6734f9d"},"schema_version":"1.0","source":{"id":"1307.4803","kind":"arxiv","version":1}},"canonical_sha256":"22b1d034407f8b4eb74c7927d51675af07fe0a50fca2b7fbb20d843c90be98b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"22b1d034407f8b4eb74c7927d51675af07fe0a50fca2b7fbb20d843c90be98b4","first_computed_at":"2026-05-18T03:18:08.633988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:08.633988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jF/wxFVeX3ZtsDTzL7G5GcmC1TneTlztnOtV8QmdLVoBr5gBOKYG1GzCBctplOo4/F7azxpbz2jwnWpdX8ZiBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:08.634652Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.4803","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6ff336d20cb6a41f711f4737dfeed5a9c078f3ed39009d738e496b87e118502","sha256:1995e487af6a83797f9acf81e42ab227cddde5ae13ea61a40086a9a5a1392b1f"],"state_sha256":"3dbbc67fc33c23a94b143e980042657c30c223bff0ddcac51be058152688759a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SmmAKpkVs4NL1JcZd/ShjFlwk3/nWeKGqcpzxQjKAbDqGOzriibWKC1bfNSIclEXDgdWj+c1vVrOAVecn+ayCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T21:48:11.399371Z","bundle_sha256":"373e49921e517cd0a200bd2843482980c12f9935ddfc897e5dc2722ff6170104"}}