{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:DOJSGQV6PN7RYPCWQOLE2SGPZV","short_pith_number":"pith:DOJSGQV6","canonical_record":{"source":{"id":"2009.09396","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-20T09:33:11Z","cross_cats_sorted":[],"title_canon_sha256":"d5c000688f157047aa8bcf1392bc6edf1bc5f036e3390cbd89c7ab9cf8267772","abstract_canon_sha256":"708aca92333de34abda79eb3a325c3c22d22e83426573e0352cb9bf3ed63caae"},"schema_version":"1.0"},"canonical_sha256":"1b932342be7b7f1c3c5683964d48cfcd5709df47ea9e10908c97883025a6cb10","source":{"kind":"arxiv","id":"2009.09396","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.09396","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"2009.09396v2","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.09396","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"DOJSGQV6PN7R","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"pith_short_16","alias_value":"DOJSGQV6PN7RYPCW","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"pith_short_8","alias_value":"DOJSGQV6","created_at":"2026-07-05T01:41:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:DOJSGQV6PN7RYPCWQOLE2SGPZV","target":"record","payload":{"canonical_record":{"source":{"id":"2009.09396","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-20T09:33:11Z","cross_cats_sorted":[],"title_canon_sha256":"d5c000688f157047aa8bcf1392bc6edf1bc5f036e3390cbd89c7ab9cf8267772","abstract_canon_sha256":"708aca92333de34abda79eb3a325c3c22d22e83426573e0352cb9bf3ed63caae"},"schema_version":"1.0"},"canonical_sha256":"1b932342be7b7f1c3c5683964d48cfcd5709df47ea9e10908c97883025a6cb10","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:41:09.631880Z","signature_b64":"fHnEbX5piFCT8FQaYpwG/r6EBQr1zWFoWobmCmvL/pJs4fAj4+Bntq/xhF+/eK1pharE9YobL2ib6yYcTdvNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b932342be7b7f1c3c5683964d48cfcd5709df47ea9e10908c97883025a6cb10","last_reissued_at":"2026-07-05T01:41:09.631478Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:41:09.631478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2009.09396","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":"2026-07-05T01:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bVJueqcDqQi4aMaeOZJ6L9YOShkiK0tLNDDnFS9AeRvgruW4woRXNfTGFJeSy7dTgHVMO9WojdKzl5e79NcgBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T14:04:47.970650Z"},"content_sha256":"6c0d73ef233c094203a73b290ad4f3fa9659f49d982ac6e6d017018892b2b79b","schema_version":"1.0","event_id":"sha256:6c0d73ef233c094203a73b290ad4f3fa9659f49d982ac6e6d017018892b2b79b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:DOJSGQV6PN7RYPCWQOLE2SGPZV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On non-empty cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chao Shi, Jianguo Qian, Peter Frankl","submitted_at":"2020-09-20T09:33:11Z","abstract_excerpt":"Let $2^{[n]}$ and $\\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\\{1,2,\\cdots,n\\}$, respectively. We call two families $\\mathscr{A}$ and $\\mathscr{B}$ cross-intersecting if $A\\cap B\\neq \\emptyset$ for any $A\\in \\mathscr{A}$ and $B\\in \\mathscr{B}$. In this paper we show that, for $n\\geq k+l,l\\geq r\\geq 1,c>0$ and $\\mathscr{A}\\subseteq \\binom{[n]}{k},\\mathscr{B}\\subseteq \\binom{[n]}{l}$, if $\\mathscr{A}$ and $\\mathscr{B}$ are cross-intersecting and $\\binom{n-r}{l-r}\\leq|\\mathscr{B}|\\leq \\binom{n-1}{l-1}$, then $$|\\mathscr{A}|+c|\\mathscr{B}|\\leq \\max\\left\\{\\binom{n}{k}-\\bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.09396","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/2009.09396/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":"2026-07-05T01:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nSioxjHcOh8v+29ynBSXnd7Ju467gPuN4JykKFmMv1lLn6K69apqtf6oBKGuxauNTWgw9A3kVkdZKDhYirP+DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T14:04:47.971027Z"},"content_sha256":"de3e9a55ccef780e1d0645e094f57ace881dde4ae78f27f7a3f931dfeda61b20","schema_version":"1.0","event_id":"sha256:de3e9a55ccef780e1d0645e094f57ace881dde4ae78f27f7a3f931dfeda61b20"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DOJSGQV6PN7RYPCWQOLE2SGPZV/bundle.json","state_url":"https://pith.science/pith/DOJSGQV6PN7RYPCWQOLE2SGPZV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DOJSGQV6PN7RYPCWQOLE2SGPZV/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":"2026-07-07T14:04:47Z","links":{"resolver":"https://pith.science/pith/DOJSGQV6PN7RYPCWQOLE2SGPZV","bundle":"https://pith.science/pith/DOJSGQV6PN7RYPCWQOLE2SGPZV/bundle.json","state":"https://pith.science/pith/DOJSGQV6PN7RYPCWQOLE2SGPZV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DOJSGQV6PN7RYPCWQOLE2SGPZV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:DOJSGQV6PN7RYPCWQOLE2SGPZV","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":"708aca92333de34abda79eb3a325c3c22d22e83426573e0352cb9bf3ed63caae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-20T09:33:11Z","title_canon_sha256":"d5c000688f157047aa8bcf1392bc6edf1bc5f036e3390cbd89c7ab9cf8267772"},"schema_version":"1.0","source":{"id":"2009.09396","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.09396","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"2009.09396v2","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.09396","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"DOJSGQV6PN7R","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"pith_short_16","alias_value":"DOJSGQV6PN7RYPCW","created_at":"2026-07-05T01:41:09Z"},{"alias_kind":"pith_short_8","alias_value":"DOJSGQV6","created_at":"2026-07-05T01:41:09Z"}],"graph_snapshots":[{"event_id":"sha256:de3e9a55ccef780e1d0645e094f57ace881dde4ae78f27f7a3f931dfeda61b20","target":"graph","created_at":"2026-07-05T01:41:09Z","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/2009.09396/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $2^{[n]}$ and $\\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\\{1,2,\\cdots,n\\}$, respectively. We call two families $\\mathscr{A}$ and $\\mathscr{B}$ cross-intersecting if $A\\cap B\\neq \\emptyset$ for any $A\\in \\mathscr{A}$ and $B\\in \\mathscr{B}$. In this paper we show that, for $n\\geq k+l,l\\geq r\\geq 1,c>0$ and $\\mathscr{A}\\subseteq \\binom{[n]}{k},\\mathscr{B}\\subseteq \\binom{[n]}{l}$, if $\\mathscr{A}$ and $\\mathscr{B}$ are cross-intersecting and $\\binom{n-r}{l-r}\\leq|\\mathscr{B}|\\leq \\binom{n-1}{l-1}$, then $$|\\mathscr{A}|+c|\\mathscr{B}|\\leq \\max\\left\\{\\binom{n}{k}-\\bi","authors_text":"Chao Shi, Jianguo Qian, Peter Frankl","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-20T09:33:11Z","title":"On non-empty cross-intersecting families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.09396","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":"sha256:6c0d73ef233c094203a73b290ad4f3fa9659f49d982ac6e6d017018892b2b79b","target":"record","created_at":"2026-07-05T01:41:09Z","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":"708aca92333de34abda79eb3a325c3c22d22e83426573e0352cb9bf3ed63caae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-20T09:33:11Z","title_canon_sha256":"d5c000688f157047aa8bcf1392bc6edf1bc5f036e3390cbd89c7ab9cf8267772"},"schema_version":"1.0","source":{"id":"2009.09396","kind":"arxiv","version":2}},"canonical_sha256":"1b932342be7b7f1c3c5683964d48cfcd5709df47ea9e10908c97883025a6cb10","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b932342be7b7f1c3c5683964d48cfcd5709df47ea9e10908c97883025a6cb10","first_computed_at":"2026-07-05T01:41:09.631478Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:41:09.631478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fHnEbX5piFCT8FQaYpwG/r6EBQr1zWFoWobmCmvL/pJs4fAj4+Bntq/xhF+/eK1pharE9YobL2ib6yYcTdvNAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:41:09.631880Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.09396","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c0d73ef233c094203a73b290ad4f3fa9659f49d982ac6e6d017018892b2b79b","sha256:de3e9a55ccef780e1d0645e094f57ace881dde4ae78f27f7a3f931dfeda61b20"],"state_sha256":"c98a414e12cfb353d8acdf0e99dab4b606f42df61811af21ac0ecb89c3c07ce2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QRXaguq8rUnvRQkKInsfhYJmtSd8aNl3YdfevnUfOws3COVvvWZqqQqCrtdKdiT4XfURP7rDG+PTIyvDE1nkBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T14:04:47.972996Z","bundle_sha256":"cd1d7068dbecdbdc639e3f29cf140de262d04f8aa694e65d10fd823363eb1ae3"}}