{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:ZNS23CT3A4SPDWTYCOYUDU2F7Z","short_pith_number":"pith:ZNS23CT3","canonical_record":{"source":{"id":"0908.1505","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-08-11T14:07:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bb149ea88424370323e87c6916c69417ff708191f21c1dfd5ffa062eb3664331","abstract_canon_sha256":"bf97e5a8922e733dab54c64495564cdf54cf4fb1cde89691b24ff1fdfee0b13f"},"schema_version":"1.0"},"canonical_sha256":"cb65ad8a7b0724f1da7813b141d345fe430c9fb040bb3e9d6e0b962d4e5fca2d","source":{"kind":"arxiv","id":"0908.1505","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.1505","created_at":"2026-05-18T04:37:22Z"},{"alias_kind":"arxiv_version","alias_value":"0908.1505v2","created_at":"2026-05-18T04:37:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.1505","created_at":"2026-05-18T04:37:22Z"},{"alias_kind":"pith_short_12","alias_value":"ZNS23CT3A4SP","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZNS23CT3A4SPDWTY","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZNS23CT3","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:ZNS23CT3A4SPDWTYCOYUDU2F7Z","target":"record","payload":{"canonical_record":{"source":{"id":"0908.1505","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-08-11T14:07:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bb149ea88424370323e87c6916c69417ff708191f21c1dfd5ffa062eb3664331","abstract_canon_sha256":"bf97e5a8922e733dab54c64495564cdf54cf4fb1cde89691b24ff1fdfee0b13f"},"schema_version":"1.0"},"canonical_sha256":"cb65ad8a7b0724f1da7813b141d345fe430c9fb040bb3e9d6e0b962d4e5fca2d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:37:22.297265Z","signature_b64":"t798QSyFMpNN9xMuH+Et/XbmGEZfAp5qK5k2mCSWjrW6LivqPg2IBx3TROIOf8XPmqzgcW4qJ/gRiqJx1Dp/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb65ad8a7b0724f1da7813b141d345fe430c9fb040bb3e9d6e0b962d4e5fca2d","last_reissued_at":"2026-05-18T04:37:22.296693Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:37:22.296693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0908.1505","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:37:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bU0zF0l86wzU0JoBfKlbwGTHtURs7QZI5AJU3CwR+gDXtMUNdH0iyWZr3zSCwTAbWqLQS/xkFrKJhcQGB2lVAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:18:43.374756Z"},"content_sha256":"1147de8ed51ba3d70a5bb443a9e79ee673e56d367ef4f51b684df67936f3bc36","schema_version":"1.0","event_id":"sha256:1147de8ed51ba3d70a5bb443a9e79ee673e56d367ef4f51b684df67936f3bc36"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:ZNS23CT3A4SPDWTYCOYUDU2F7Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Adam Van Tuyl, Christopher A. Francisco, Huy Tai Ha","submitted_at":"2009-08-11T14:07:06Z","abstract_excerpt":"There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1505","kind":"arxiv","version":2},"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":"2026-05-18T04:37:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bvy2NtsSjJ36vlCbUINr1F3jPiWX0zX1VQSsVjKDe/5sY4qRSUBL3MFL8We+/kySw70esMhsN643SAhH/31rBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:18:43.375134Z"},"content_sha256":"fdc4050e0ab281f199c18cdc623f8a8f7d29656df7bc8e3cc7ffd2c18d6a0ff6","schema_version":"1.0","event_id":"sha256:fdc4050e0ab281f199c18cdc623f8a8f7d29656df7bc8e3cc7ffd2c18d6a0ff6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZNS23CT3A4SPDWTYCOYUDU2F7Z/bundle.json","state_url":"https://pith.science/pith/ZNS23CT3A4SPDWTYCOYUDU2F7Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZNS23CT3A4SPDWTYCOYUDU2F7Z/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":"2026-05-28T05:18:43Z","links":{"resolver":"https://pith.science/pith/ZNS23CT3A4SPDWTYCOYUDU2F7Z","bundle":"https://pith.science/pith/ZNS23CT3A4SPDWTYCOYUDU2F7Z/bundle.json","state":"https://pith.science/pith/ZNS23CT3A4SPDWTYCOYUDU2F7Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZNS23CT3A4SPDWTYCOYUDU2F7Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ZNS23CT3A4SPDWTYCOYUDU2F7Z","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":"bf97e5a8922e733dab54c64495564cdf54cf4fb1cde89691b24ff1fdfee0b13f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-08-11T14:07:06Z","title_canon_sha256":"bb149ea88424370323e87c6916c69417ff708191f21c1dfd5ffa062eb3664331"},"schema_version":"1.0","source":{"id":"0908.1505","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.1505","created_at":"2026-05-18T04:37:22Z"},{"alias_kind":"arxiv_version","alias_value":"0908.1505v2","created_at":"2026-05-18T04:37:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.1505","created_at":"2026-05-18T04:37:22Z"},{"alias_kind":"pith_short_12","alias_value":"ZNS23CT3A4SP","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZNS23CT3A4SPDWTY","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZNS23CT3","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:fdc4050e0ab281f199c18cdc623f8a8f7d29656df7bc8e3cc7ffd2c18d6a0ff6","target":"graph","created_at":"2026-05-18T04:37:22Z","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":"There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely al","authors_text":"Adam Van Tuyl, Christopher A. Francisco, Huy Tai Ha","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-08-11T14:07:06Z","title":"Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1505","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":"sha256:1147de8ed51ba3d70a5bb443a9e79ee673e56d367ef4f51b684df67936f3bc36","target":"record","created_at":"2026-05-18T04:37:22Z","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":"bf97e5a8922e733dab54c64495564cdf54cf4fb1cde89691b24ff1fdfee0b13f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-08-11T14:07:06Z","title_canon_sha256":"bb149ea88424370323e87c6916c69417ff708191f21c1dfd5ffa062eb3664331"},"schema_version":"1.0","source":{"id":"0908.1505","kind":"arxiv","version":2}},"canonical_sha256":"cb65ad8a7b0724f1da7813b141d345fe430c9fb040bb3e9d6e0b962d4e5fca2d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb65ad8a7b0724f1da7813b141d345fe430c9fb040bb3e9d6e0b962d4e5fca2d","first_computed_at":"2026-05-18T04:37:22.296693Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:37:22.296693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t798QSyFMpNN9xMuH+Et/XbmGEZfAp5qK5k2mCSWjrW6LivqPg2IBx3TROIOf8XPmqzgcW4qJ/gRiqJx1Dp/Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:37:22.297265Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.1505","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1147de8ed51ba3d70a5bb443a9e79ee673e56d367ef4f51b684df67936f3bc36","sha256:fdc4050e0ab281f199c18cdc623f8a8f7d29656df7bc8e3cc7ffd2c18d6a0ff6"],"state_sha256":"e2b1679eaa5ec7dd07fb221a2e0043f13ea4013ffbad3595cf063e13cf44c58f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hggRheBUneu27v/R3wU3XO/nU81LB/y33nEpj7jw+6kif/BtAVvj5oeGPIs/FLBzGw9eC3Iwn9KW3UgRSKw9Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T05:18:43.377634Z","bundle_sha256":"9b201a081edce05c91e35ac62016a9ade5769fe683afaa6951e626315faea002"}}