{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:UK5J5HUCUQB3DRNO7LHBKP7T3N","short_pith_number":"pith:UK5J5HUC","canonical_record":{"source":{"id":"2306.03473","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-06-06T07:47:05Z","cross_cats_sorted":[],"title_canon_sha256":"f7c7c648baaae7e1a5dc7d80f45f39b700207488f631c73307fd636580cdd216","abstract_canon_sha256":"083dd3dc9619f10b063ea0bccdcce36b1b46f5b88b60f6734e8ce15fb725ef22"},"schema_version":"1.0"},"canonical_sha256":"a2ba9e9e82a403b1c5aeface153ff3db609c18e40f00e82ab4fc24310e7eba76","source":{"kind":"arxiv","id":"2306.03473","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.03473","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"arxiv_version","alias_value":"2306.03473v2","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.03473","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"pith_short_12","alias_value":"UK5J5HUCUQB3","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"pith_short_16","alias_value":"UK5J5HUCUQB3DRNO","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"pith_short_8","alias_value":"UK5J5HUC","created_at":"2026-07-05T07:46:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:UK5J5HUCUQB3DRNO7LHBKP7T3N","target":"record","payload":{"canonical_record":{"source":{"id":"2306.03473","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-06-06T07:47:05Z","cross_cats_sorted":[],"title_canon_sha256":"f7c7c648baaae7e1a5dc7d80f45f39b700207488f631c73307fd636580cdd216","abstract_canon_sha256":"083dd3dc9619f10b063ea0bccdcce36b1b46f5b88b60f6734e8ce15fb725ef22"},"schema_version":"1.0"},"canonical_sha256":"a2ba9e9e82a403b1c5aeface153ff3db609c18e40f00e82ab4fc24310e7eba76","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:46:19.348192Z","signature_b64":"zghFWwqSsVw/6dsOZRcv7F6JQk3qL0yrEB3YJmGO0w9r1Xi15i07AohJ3XdNtbfxmQ1EZVogTPxaFODZTwOFCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2ba9e9e82a403b1c5aeface153ff3db609c18e40f00e82ab4fc24310e7eba76","last_reissued_at":"2026-07-05T07:46:19.347701Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:46:19.347701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2306.03473","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":"2026-07-05T07:46:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"awCmgeKRUN75ZT9cbUHwoTh3igeb8DtF7l9vxuvOu+OjSzgamvGuO+flFM8WvSnWysqMv23026z81vaX1LVoCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T03:19:41.707588Z"},"content_sha256":"1664a6779211182532086e7327c67a8ad0ff740913b3e4465db2b9130e14d554","schema_version":"1.0","event_id":"sha256:1664a6779211182532086e7327c67a8ad0ff740913b3e4465db2b9130e14d554"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:UK5J5HUCUQB3DRNO7LHBKP7T3N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-empty pairwise cross-intersecting families","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yang Huang, Yuejian Peng","submitted_at":"2023-06-06T07:47:05Z","abstract_excerpt":"Two families $\\mathcal{A}$ and $\\mathcal{B}$ are cross-intersecting if $A\\cap B\\ne \\emptyset$ for any $A\\in \\mathcal{A}$ and $B\\in \\mathcal{B}$. We call $t$ families $\\mathcal{A}_1, \\mathcal{A}_2,\\dots, \\mathcal{A}_t$ pairwise cross-intersecting families if $\\mathcal{A}_i$ and $\\mathcal{A}_j$ are cross-intersecting when $1\\le i<j \\le t$. Additionally, if $\\mathcal{A}_j\\ne \\emptyset$ for each $j\\in [t]$, then we say that $\\mathcal{A}_1, \\mathcal{A}_2,\\dots, \\mathcal{A}_t$ are non-empty pairwise cross-intersecting. Let $\\mathcal{A}_1\\subset{[n]\\choose k_1}, \\mathcal{A}_2\\subset{[n]\\choose k_2}, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.03473","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/2306.03473/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":"2026-07-05T07:46:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PLj8tSdUsInOhgEH8t0MWdoENwhTEmcbx8urNrovFf/kusMbCkKgBP+IHzjdRHhRaRm1XONJ1YTAa0SS/MuvCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T03:19:41.708241Z"},"content_sha256":"2fbe81bb4e2cd4a0a5dd25d77c58218a4aeda1b04efd8d6b2ec4e5287d8ddbd6","schema_version":"1.0","event_id":"sha256:2fbe81bb4e2cd4a0a5dd25d77c58218a4aeda1b04efd8d6b2ec4e5287d8ddbd6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UK5J5HUCUQB3DRNO7LHBKP7T3N/bundle.json","state_url":"https://pith.science/pith/UK5J5HUCUQB3DRNO7LHBKP7T3N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UK5J5HUCUQB3DRNO7LHBKP7T3N/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":"2026-08-18T03:19:41Z","links":{"resolver":"https://pith.science/pith/UK5J5HUCUQB3DRNO7LHBKP7T3N","bundle":"https://pith.science/pith/UK5J5HUCUQB3DRNO7LHBKP7T3N/bundle.json","state":"https://pith.science/pith/UK5J5HUCUQB3DRNO7LHBKP7T3N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UK5J5HUCUQB3DRNO7LHBKP7T3N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:UK5J5HUCUQB3DRNO7LHBKP7T3N","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":"083dd3dc9619f10b063ea0bccdcce36b1b46f5b88b60f6734e8ce15fb725ef22","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-06-06T07:47:05Z","title_canon_sha256":"f7c7c648baaae7e1a5dc7d80f45f39b700207488f631c73307fd636580cdd216"},"schema_version":"1.0","source":{"id":"2306.03473","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.03473","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"arxiv_version","alias_value":"2306.03473v2","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.03473","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"pith_short_12","alias_value":"UK5J5HUCUQB3","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"pith_short_16","alias_value":"UK5J5HUCUQB3DRNO","created_at":"2026-07-05T07:46:19Z"},{"alias_kind":"pith_short_8","alias_value":"UK5J5HUC","created_at":"2026-07-05T07:46:19Z"}],"graph_snapshots":[{"event_id":"sha256:2fbe81bb4e2cd4a0a5dd25d77c58218a4aeda1b04efd8d6b2ec4e5287d8ddbd6","target":"graph","created_at":"2026-07-05T07:46:19Z","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/2306.03473/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Two families $\\mathcal{A}$ and $\\mathcal{B}$ are cross-intersecting if $A\\cap B\\ne \\emptyset$ for any $A\\in \\mathcal{A}$ and $B\\in \\mathcal{B}$. We call $t$ families $\\mathcal{A}_1, \\mathcal{A}_2,\\dots, \\mathcal{A}_t$ pairwise cross-intersecting families if $\\mathcal{A}_i$ and $\\mathcal{A}_j$ are cross-intersecting when $1\\le i<j \\le t$. Additionally, if $\\mathcal{A}_j\\ne \\emptyset$ for each $j\\in [t]$, then we say that $\\mathcal{A}_1, \\mathcal{A}_2,\\dots, \\mathcal{A}_t$ are non-empty pairwise cross-intersecting. Let $\\mathcal{A}_1\\subset{[n]\\choose k_1}, \\mathcal{A}_2\\subset{[n]\\choose k_2}, ","authors_text":"Yang Huang, Yuejian Peng","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-06-06T07:47:05Z","title":"Non-empty pairwise cross-intersecting families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.03473","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":"sha256:1664a6779211182532086e7327c67a8ad0ff740913b3e4465db2b9130e14d554","target":"record","created_at":"2026-07-05T07:46:19Z","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":"083dd3dc9619f10b063ea0bccdcce36b1b46f5b88b60f6734e8ce15fb725ef22","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-06-06T07:47:05Z","title_canon_sha256":"f7c7c648baaae7e1a5dc7d80f45f39b700207488f631c73307fd636580cdd216"},"schema_version":"1.0","source":{"id":"2306.03473","kind":"arxiv","version":2}},"canonical_sha256":"a2ba9e9e82a403b1c5aeface153ff3db609c18e40f00e82ab4fc24310e7eba76","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a2ba9e9e82a403b1c5aeface153ff3db609c18e40f00e82ab4fc24310e7eba76","first_computed_at":"2026-07-05T07:46:19.347701Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:46:19.347701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zghFWwqSsVw/6dsOZRcv7F6JQk3qL0yrEB3YJmGO0w9r1Xi15i07AohJ3XdNtbfxmQ1EZVogTPxaFODZTwOFCw==","signature_status":"signed_v1","signed_at":"2026-07-05T07:46:19.348192Z","signed_message":"canonical_sha256_bytes"},"source_id":"2306.03473","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1664a6779211182532086e7327c67a8ad0ff740913b3e4465db2b9130e14d554","sha256:2fbe81bb4e2cd4a0a5dd25d77c58218a4aeda1b04efd8d6b2ec4e5287d8ddbd6"],"state_sha256":"c25f2312add958f0b252230799dd9fe10e85612ad8c2ed7f8d0cccb46bb63928"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"enXu7P07A6YQyt925+rI8du9uLToL+IRH0rua+hhsiFKMBOtjiYczaAu1Lql9jYs4k26HaSKFo2OPJ/IfeIcDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T03:19:41.718073Z","bundle_sha256":"16c3e674dda21fe20c4c364187e3bfcde6fc95e4fa320cda490ccc75049c4323"}}