{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:GRJHBS4AIFQT76KZXYARB3FFFQ","short_pith_number":"pith:GRJHBS4A","canonical_record":{"source":{"id":"1109.4775","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-22T11:31:47Z","cross_cats_sorted":[],"title_canon_sha256":"3cf98384492b8ed48fcfea7d65bed98fcd8ddbdb36854a9de89605bd5febf950","abstract_canon_sha256":"7e7b0be1acd2ce2f297e4dea08fd8aa1f8b1434517bf6faf902242bb6bbabc94"},"schema_version":"1.0"},"canonical_sha256":"345270cb8041613ff959be0110eca52c2b227a066433f5d2c90bf74a3d536c2f","source":{"kind":"arxiv","id":"1109.4775","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4775","created_at":"2026-05-18T03:27:52Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4775v4","created_at":"2026-05-18T03:27:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4775","created_at":"2026-05-18T03:27:52Z"},{"alias_kind":"pith_short_12","alias_value":"GRJHBS4AIFQT","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"GRJHBS4AIFQT76KZ","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"GRJHBS4A","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:GRJHBS4AIFQT76KZXYARB3FFFQ","target":"record","payload":{"canonical_record":{"source":{"id":"1109.4775","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-22T11:31:47Z","cross_cats_sorted":[],"title_canon_sha256":"3cf98384492b8ed48fcfea7d65bed98fcd8ddbdb36854a9de89605bd5febf950","abstract_canon_sha256":"7e7b0be1acd2ce2f297e4dea08fd8aa1f8b1434517bf6faf902242bb6bbabc94"},"schema_version":"1.0"},"canonical_sha256":"345270cb8041613ff959be0110eca52c2b227a066433f5d2c90bf74a3d536c2f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:52.469926Z","signature_b64":"8Rxw6DqF5NsWgk94sVRKeesOJgB5gk0hgtArm5Ba8yijkW9YeQXSImhHBfGvirNWTVe1KBY23CEf/VGKagaWBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"345270cb8041613ff959be0110eca52c2b227a066433f5d2c90bf74a3d536c2f","last_reissued_at":"2026-05-18T03:27:52.469384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:52.469384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.4775","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:27:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZlUeIrOcw0zx2G32n9gXyQDuz0XxveE66fpHb3nhOvJ599U2nWS098dXaaSGkdcGbePMtYz/bjiohzcZwDyLDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:09:20.595161Z"},"content_sha256":"ad5c9b7e4908861822c0672afb3853409702f883224346f94350eb99dc917ff1","schema_version":"1.0","event_id":"sha256:ad5c9b7e4908861822c0672afb3853409702f883224346f94350eb99dc917ff1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:GRJHBS4AIFQT76KZXYARB3FFFQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extremal problems related to Betti numbers of flag complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michal Adamaszek","submitted_at":"2011-09-22T11:31:47Z","abstract_excerpt":"We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32^n for the sum of Betti numbers of any n-vertex flag complex and 1.25^n for the independence complex of a triangle-free graph. These findings imply upper bounds for the Betti numbers of various related classes of spaces, including the neighbourhood complex of a graph. We also make some related observations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4775","kind":"arxiv","version":4},"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":"2026-05-18T03:27:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qZ/aApGcxR9EMPFp+T8Ju4FlfemLsmdG01fyk1k3eQXScRoUSVJ/BbKQCMOx0/Dtvy/dJtYfcqdLJEhGrc7DCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:09:20.595863Z"},"content_sha256":"0d5a5f1bc24dda75ba4976b5f22c8cdf9bac846d7761970c108ad704318aa43c","schema_version":"1.0","event_id":"sha256:0d5a5f1bc24dda75ba4976b5f22c8cdf9bac846d7761970c108ad704318aa43c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GRJHBS4AIFQT76KZXYARB3FFFQ/bundle.json","state_url":"https://pith.science/pith/GRJHBS4AIFQT76KZXYARB3FFFQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GRJHBS4AIFQT76KZXYARB3FFFQ/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":"2026-05-28T20:09:20Z","links":{"resolver":"https://pith.science/pith/GRJHBS4AIFQT76KZXYARB3FFFQ","bundle":"https://pith.science/pith/GRJHBS4AIFQT76KZXYARB3FFFQ/bundle.json","state":"https://pith.science/pith/GRJHBS4AIFQT76KZXYARB3FFFQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GRJHBS4AIFQT76KZXYARB3FFFQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:GRJHBS4AIFQT76KZXYARB3FFFQ","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":"7e7b0be1acd2ce2f297e4dea08fd8aa1f8b1434517bf6faf902242bb6bbabc94","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-22T11:31:47Z","title_canon_sha256":"3cf98384492b8ed48fcfea7d65bed98fcd8ddbdb36854a9de89605bd5febf950"},"schema_version":"1.0","source":{"id":"1109.4775","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4775","created_at":"2026-05-18T03:27:52Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4775v4","created_at":"2026-05-18T03:27:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4775","created_at":"2026-05-18T03:27:52Z"},{"alias_kind":"pith_short_12","alias_value":"GRJHBS4AIFQT","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"GRJHBS4AIFQT76KZ","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"GRJHBS4A","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:0d5a5f1bc24dda75ba4976b5f22c8cdf9bac846d7761970c108ad704318aa43c","target":"graph","created_at":"2026-05-18T03:27:52Z","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":"We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32^n for the sum of Betti numbers of any n-vertex flag complex and 1.25^n for the independence complex of a triangle-free graph. These findings imply upper bounds for the Betti numbers of various related classes of spaces, including the neighbourhood complex of a graph. We also make some related observations.","authors_text":"Michal Adamaszek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-22T11:31:47Z","title":"Extremal problems related to Betti numbers of flag complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4775","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:ad5c9b7e4908861822c0672afb3853409702f883224346f94350eb99dc917ff1","target":"record","created_at":"2026-05-18T03:27:52Z","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":"7e7b0be1acd2ce2f297e4dea08fd8aa1f8b1434517bf6faf902242bb6bbabc94","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-22T11:31:47Z","title_canon_sha256":"3cf98384492b8ed48fcfea7d65bed98fcd8ddbdb36854a9de89605bd5febf950"},"schema_version":"1.0","source":{"id":"1109.4775","kind":"arxiv","version":4}},"canonical_sha256":"345270cb8041613ff959be0110eca52c2b227a066433f5d2c90bf74a3d536c2f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"345270cb8041613ff959be0110eca52c2b227a066433f5d2c90bf74a3d536c2f","first_computed_at":"2026-05-18T03:27:52.469384Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:27:52.469384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Rxw6DqF5NsWgk94sVRKeesOJgB5gk0hgtArm5Ba8yijkW9YeQXSImhHBfGvirNWTVe1KBY23CEf/VGKagaWBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:27:52.469926Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.4775","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ad5c9b7e4908861822c0672afb3853409702f883224346f94350eb99dc917ff1","sha256:0d5a5f1bc24dda75ba4976b5f22c8cdf9bac846d7761970c108ad704318aa43c"],"state_sha256":"f2c52a82fc5a594fdc51e00754f7c313c20d3aedfb01fd18a2a30a6b4d9a6fc7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dIeNdbO8qhscbdAX8DZ6TVE1GYG3FUZb2vprSAcoP+3ygFw7vDPPYrI7FnstYMY6z4HA5g5YdChgMKv8AxD0BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T20:09:20.599553Z","bundle_sha256":"5b7e45d3bd6330d84b1b2ed1e9a8a197fbe0aafb1c2b5d289b0ef980d1593f02"}}