{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WJ4RJMIQDCWLAK2YT4DRBSCM6U","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":"50478064900c5ac5c190302d0d9e368bae8605bc0b0c65d07b9ccca859b4f279","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-12T20:59:21Z","title_canon_sha256":"8dfd77af40cdd07e13a3a41cb39d500891f4b022b3d65513e0a92a70011b851b"},"schema_version":"1.0","source":{"id":"1510.03461","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03461","created_at":"2026-05-18T01:30:17Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03461v1","created_at":"2026-05-18T01:30:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03461","created_at":"2026-05-18T01:30:17Z"},{"alias_kind":"pith_short_12","alias_value":"WJ4RJMIQDCWL","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WJ4RJMIQDCWLAK2Y","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WJ4RJMIQ","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:54fe999903bec765c79ab77df6b502de001d602e3cf1fcd7c0ee57870d16b28d","target":"graph","created_at":"2026-05-18T01:30:17Z","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":"Given a family of $r$-uniform hypergraphs ${\\cal F}$ (or $r$-graphs for brevity), the Tur\\'an number $ex(n,{\\cal F})$ of ${\\cal F}$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain any member of ${\\cal F}$. A pair $\\{u,v\\}$ is covered in a hypergraph $G$ if some edge of $G$ contains $\\{u,v\\}$. Given an $r$-graph $F$ and a positive integer $p\\geq n(F)$, let $H^F_p$ denote the $r$-graph obtained as follows. Label the vertices of $F$ as $v_1,\\ldots, v_{n(F)}$. Add new vertices $v_{n(F)+1},\\ldots, v_p$. For each pair of vertices $v_i,v_j$ not covered in $F$, add","authors_text":"Axel Brandt, David Irwin, Tao Jiang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-12T20:59:21Z","title":"Stability and Tur\\'an numbers of a class of hypergraphs via Lagrangians"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03461","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":"sha256:7f5292a0ddd01dcb1bd7b8ed24578a6b3fe9c1b9e48b8a37383e23e5bb2bbb50","target":"record","created_at":"2026-05-18T01:30:17Z","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":"50478064900c5ac5c190302d0d9e368bae8605bc0b0c65d07b9ccca859b4f279","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-12T20:59:21Z","title_canon_sha256":"8dfd77af40cdd07e13a3a41cb39d500891f4b022b3d65513e0a92a70011b851b"},"schema_version":"1.0","source":{"id":"1510.03461","kind":"arxiv","version":1}},"canonical_sha256":"b27914b11018acb02b589f0710c84cf52cff73461fbfa7bb1b5c3af9dbfd4cc4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b27914b11018acb02b589f0710c84cf52cff73461fbfa7bb1b5c3af9dbfd4cc4","first_computed_at":"2026-05-18T01:30:17.154060Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:17.154060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kCIssvf++T8HLJmfAJMbzsoI2zqiiXMwjYQok8xFmSnW69smZrkjGcK8Go6QruOzeIv9JLt1snKRCx4qPaRXDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:17.154823Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.03461","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f5292a0ddd01dcb1bd7b8ed24578a6b3fe9c1b9e48b8a37383e23e5bb2bbb50","sha256:54fe999903bec765c79ab77df6b502de001d602e3cf1fcd7c0ee57870d16b28d"],"state_sha256":"e7cd60bfc6f6a240a50e67266ba2740fe34760db16ae83aa23818882746f408d"}