{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:63ZMOE2ANKT6JPLBHKAW7K43RH","short_pith_number":"pith:63ZMOE2A","canonical_record":{"source":{"id":"2102.02103","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-03T15:17:31Z","cross_cats_sorted":[],"title_canon_sha256":"b5b807010d45e3922491040926488840ee901f5821010acf08cb4ff392b41d3b","abstract_canon_sha256":"bf0ad27b3f30392af48e7ebe9f243b79da251cc48c902eb17992033f9f04f701"},"schema_version":"1.0"},"canonical_sha256":"f6f2c713406aa7e4bd613a816fab9b89d17887a20b0a258b4fe87954d807d9b1","source":{"kind":"arxiv","id":"2102.02103","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.02103","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"arxiv_version","alias_value":"2102.02103v2","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.02103","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"pith_short_12","alias_value":"63ZMOE2ANKT6","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"pith_short_16","alias_value":"63ZMOE2ANKT6JPLB","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"pith_short_8","alias_value":"63ZMOE2A","created_at":"2026-07-05T02:15:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:63ZMOE2ANKT6JPLBHKAW7K43RH","target":"record","payload":{"canonical_record":{"source":{"id":"2102.02103","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-03T15:17:31Z","cross_cats_sorted":[],"title_canon_sha256":"b5b807010d45e3922491040926488840ee901f5821010acf08cb4ff392b41d3b","abstract_canon_sha256":"bf0ad27b3f30392af48e7ebe9f243b79da251cc48c902eb17992033f9f04f701"},"schema_version":"1.0"},"canonical_sha256":"f6f2c713406aa7e4bd613a816fab9b89d17887a20b0a258b4fe87954d807d9b1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:15:41.789465Z","signature_b64":"0NQSF0zFmjnd1qG7jU6aek0pF8Rg6qoicRxXWR2MImK55xwyKo6ltHZS/Ri8J7EH3ZGYQe8choqhm1NUCaqBCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6f2c713406aa7e4bd613a816fab9b89d17887a20b0a258b4fe87954d807d9b1","last_reissued_at":"2026-07-05T02:15:41.789022Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:15:41.789022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2102.02103","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":"2026-07-05T02:15:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hovQVmoRXlTNoOG7x4bKOMX24o4osGu/lgPoqdg1Ng6cKN5g0J8AXhtmDoxDG43Uun3PnouchfhYBGdIFp+YBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T18:41:16.547850Z"},"content_sha256":"b39ac11062d3001af2e5e9a7a42b7cd9ec59d008fc3ad3d06dc031fb48b4d127","schema_version":"1.0","event_id":"sha256:b39ac11062d3001af2e5e9a7a42b7cd9ec59d008fc3ad3d06dc031fb48b4d127"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:63ZMOE2ANKT6JPLBHKAW7K43RH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hypergraphs with many extremal configurations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher, Dhruv Mubayi, Xizhi Liu","submitted_at":"2021-02-03T15:17:31Z","abstract_excerpt":"For every positive integer $t$ we construct a finite family of triple systems ${\\mathcal M}_t$, determine its Tur\\'{a}n number, and show that there are $t$ extremal ${\\mathcal M}_t$-free configurations that are far from each other in edit-distance. We also prove a strong stability theorem: every ${\\mathcal M}_t$-free triple system whose size is close to the maximum size is a subgraph of one of these $t$ extremal configurations after removing a small proportion of vertices. This is the first stability theorem for a hypergraph problem with an arbitrary (finite) number of extremal configurations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.02103","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2102.02103/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:15:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HoSDqwzps73hr25yMDbZ7PDccntpKU4+H8pFl7umnBiYEv9YP0gFoCuh+9syE9eedv5sZpCL0334zMk/UPjRDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T18:41:16.548395Z"},"content_sha256":"c837ee807e6a235c40d8e74b98f978d3d083035da0f81db49822c8e64a3dd80b","schema_version":"1.0","event_id":"sha256:c837ee807e6a235c40d8e74b98f978d3d083035da0f81db49822c8e64a3dd80b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/bundle.json","state_url":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/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":"2026-08-08T18:41:16Z","links":{"resolver":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH","bundle":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/bundle.json","state":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:63ZMOE2ANKT6JPLBHKAW7K43RH","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":"bf0ad27b3f30392af48e7ebe9f243b79da251cc48c902eb17992033f9f04f701","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-03T15:17:31Z","title_canon_sha256":"b5b807010d45e3922491040926488840ee901f5821010acf08cb4ff392b41d3b"},"schema_version":"1.0","source":{"id":"2102.02103","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.02103","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"arxiv_version","alias_value":"2102.02103v2","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.02103","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"pith_short_12","alias_value":"63ZMOE2ANKT6","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"pith_short_16","alias_value":"63ZMOE2ANKT6JPLB","created_at":"2026-07-05T02:15:41Z"},{"alias_kind":"pith_short_8","alias_value":"63ZMOE2A","created_at":"2026-07-05T02:15:41Z"}],"graph_snapshots":[{"event_id":"sha256:c837ee807e6a235c40d8e74b98f978d3d083035da0f81db49822c8e64a3dd80b","target":"graph","created_at":"2026-07-05T02:15:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2102.02103/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For every positive integer $t$ we construct a finite family of triple systems ${\\mathcal M}_t$, determine its Tur\\'{a}n number, and show that there are $t$ extremal ${\\mathcal M}_t$-free configurations that are far from each other in edit-distance. We also prove a strong stability theorem: every ${\\mathcal M}_t$-free triple system whose size is close to the maximum size is a subgraph of one of these $t$ extremal configurations after removing a small proportion of vertices. This is the first stability theorem for a hypergraph problem with an arbitrary (finite) number of extremal configurations.","authors_text":"Christian Reiher, Dhruv Mubayi, Xizhi Liu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-03T15:17:31Z","title":"Hypergraphs with many extremal configurations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.02103","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:b39ac11062d3001af2e5e9a7a42b7cd9ec59d008fc3ad3d06dc031fb48b4d127","target":"record","created_at":"2026-07-05T02:15:41Z","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":"bf0ad27b3f30392af48e7ebe9f243b79da251cc48c902eb17992033f9f04f701","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-03T15:17:31Z","title_canon_sha256":"b5b807010d45e3922491040926488840ee901f5821010acf08cb4ff392b41d3b"},"schema_version":"1.0","source":{"id":"2102.02103","kind":"arxiv","version":2}},"canonical_sha256":"f6f2c713406aa7e4bd613a816fab9b89d17887a20b0a258b4fe87954d807d9b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f6f2c713406aa7e4bd613a816fab9b89d17887a20b0a258b4fe87954d807d9b1","first_computed_at":"2026-07-05T02:15:41.789022Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:15:41.789022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0NQSF0zFmjnd1qG7jU6aek0pF8Rg6qoicRxXWR2MImK55xwyKo6ltHZS/Ri8J7EH3ZGYQe8choqhm1NUCaqBCw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:15:41.789465Z","signed_message":"canonical_sha256_bytes"},"source_id":"2102.02103","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b39ac11062d3001af2e5e9a7a42b7cd9ec59d008fc3ad3d06dc031fb48b4d127","sha256:c837ee807e6a235c40d8e74b98f978d3d083035da0f81db49822c8e64a3dd80b"],"state_sha256":"963560bc484cefaf5f405d77afab6ddebe1b4b5e1082b23274cf5e99c6cd2383"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4LmP0HQZze/xG5Qih/OPcWD9vkNYVt03YXs73ZUmKlgJ1eWMxhs72ucK2pSGGFlkOQBnALOgSTGf04jYA6c7CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T18:41:16.552798Z","bundle_sha256":"bf94156b89f2056938913be2e89e8acceb9d05180f1c89ec7e69d2f37a12d441"}}