{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:GP5UK4PFZKN2CWAKF6WUPYZ2AM","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":"819cab4a1de6661c7c0de09a934713594f9564964b90e06dd609e7fdced4972b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-05-27T20:20:29Z","title_canon_sha256":"b72d6c9df46355d7761b730b3c0e018d26fa43bebe76c738093b3928ea6157a8"},"schema_version":"1.0","source":{"id":"2405.17639","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.17639","created_at":"2026-07-05T08:38:09Z"},{"alias_kind":"arxiv_version","alias_value":"2405.17639v2","created_at":"2026-07-05T08:38:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.17639","created_at":"2026-07-05T08:38:09Z"},{"alias_kind":"pith_short_12","alias_value":"GP5UK4PFZKN2","created_at":"2026-07-05T08:38:09Z"},{"alias_kind":"pith_short_16","alias_value":"GP5UK4PFZKN2CWAK","created_at":"2026-07-05T08:38:09Z"},{"alias_kind":"pith_short_8","alias_value":"GP5UK4PF","created_at":"2026-07-05T08:38:09Z"}],"graph_snapshots":[{"event_id":"sha256:59fb79f61b3b27d448aca05aefd3e2d97694a7639baa24f91dc0cf4d69b43917","target":"graph","created_at":"2026-07-05T08:38:09Z","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/2405.17639/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A family of disjoint pairs of finite sets $\\mathcal{P}=\\{(A_i,B_i)\\mid i\\in[m]\\}$ is called a Bollob\\'as system if $A_i\\cap B_j\\neq\\emptyset$ for every $i\\neq j$, and a skew Bollob\\'as system if $A_i\\cap B_j\\neq\\emptyset$ for every $i<j$. Bollob\\'as proved that for a Bollob\\'as system, the inequality \\begin{equation*} \\sum_{i=1}^m\\binom{|A_i|+|B_i|}{|A_i|}^{-1}\\leq 1 \\end{equation*} holds. Heged\\\"{u}s and Frankl generalized this theorem to skew Bollob\\'as systems with the inequality \\begin{equation*} \\sum_{i=1}^m\\binom{|A_i|+|B_i|}{|A_i|}^{-1}\\leq 1+n, \\end{equation*} provided $A_i,B_i\\subsete","authors_text":"Erfei Yue","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-05-27T20:20:29Z","title":"Some new Bollob\\'as-type inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.17639","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":"sha256:038bd5537359cb3990ac9ad05eb57d20a4d113f416d0eb39f71c57e1243c08d4","target":"record","created_at":"2026-07-05T08:38:09Z","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":"819cab4a1de6661c7c0de09a934713594f9564964b90e06dd609e7fdced4972b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-05-27T20:20:29Z","title_canon_sha256":"b72d6c9df46355d7761b730b3c0e018d26fa43bebe76c738093b3928ea6157a8"},"schema_version":"1.0","source":{"id":"2405.17639","kind":"arxiv","version":2}},"canonical_sha256":"33fb4571e5ca9ba1580a2fad47e33a030de866a64a9acf72419c69495a569fae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33fb4571e5ca9ba1580a2fad47e33a030de866a64a9acf72419c69495a569fae","first_computed_at":"2026-07-05T08:38:09.976147Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:38:09.976147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5QZxxLBeRVEKc1oavrUBNxzB06aPZWJAJiCPzVrUa3qMvT1A8btCSChN5y2stSRMvUeqtqbhvjBrGu3bJtf/Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:38:09.976593Z","signed_message":"canonical_sha256_bytes"},"source_id":"2405.17639","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:038bd5537359cb3990ac9ad05eb57d20a4d113f416d0eb39f71c57e1243c08d4","sha256:59fb79f61b3b27d448aca05aefd3e2d97694a7639baa24f91dc0cf4d69b43917"],"state_sha256":"a1f7b03b7ea09a96b55d4ca84f4b675b85de98d5b4ea90e0ea1847ce613d517b"}