{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:63ZMOE2ANKT6JPLBHKAW7K43RH","short_pith_number":"pith:63ZMOE2A","schema_version":"1.0","canonical_sha256":"f6f2c713406aa7e4bd613a816fab9b89d17887a20b0a258b4fe87954d807d9b1","source":{"kind":"arxiv","id":"2102.02103","version":2},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2102.02103","created_at":"2026-07-05T02:15:41.789075+00:00"},{"alias_kind":"arxiv_version","alias_value":"2102.02103v2","created_at":"2026-07-05T02:15:41.789075+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.02103","created_at":"2026-07-05T02:15:41.789075+00:00"},{"alias_kind":"pith_short_12","alias_value":"63ZMOE2ANKT6","created_at":"2026-07-05T02:15:41.789075+00:00"},{"alias_kind":"pith_short_16","alias_value":"63ZMOE2ANKT6JPLB","created_at":"2026-07-05T02:15:41.789075+00:00"},{"alias_kind":"pith_short_8","alias_value":"63ZMOE2A","created_at":"2026-07-05T02:15:41.789075+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.03223","citing_title":"The Tur\\'{a}n density of short tight cycles","ref_index":17,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH","json":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH.json","graph_json":"https://pith.science/api/pith-number/63ZMOE2ANKT6JPLBHKAW7K43RH/graph.json","events_json":"https://pith.science/api/pith-number/63ZMOE2ANKT6JPLBHKAW7K43RH/events.json","paper":"https://pith.science/paper/63ZMOE2A"},"agent_actions":{"view_html":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH","download_json":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH.json","view_paper":"https://pith.science/paper/63ZMOE2A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2102.02103&json=true","fetch_graph":"https://pith.science/api/pith-number/63ZMOE2ANKT6JPLBHKAW7K43RH/graph.json","fetch_events":"https://pith.science/api/pith-number/63ZMOE2ANKT6JPLBHKAW7K43RH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/action/storage_attestation","attest_author":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/action/author_attestation","sign_citation":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/action/citation_signature","submit_replication":"https://pith.science/pith/63ZMOE2ANKT6JPLBHKAW7K43RH/action/replication_record"}},"created_at":"2026-07-05T02:15:41.789075+00:00","updated_at":"2026-07-05T02:15:41.789075+00:00"}