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We prove the sharp inequality $$\\prod_{i=1}^r\\frac{|\\mathcal{F}_i|}{\\binom{n}{k_i}}\\leq \\prod_{i=1}^r\\frac{k_i}{n},$$ with equality attained by the corresponding levels of a common $1$-star. As a consequence, we obtain the analogous $p_i$-biased measure theorem for $0\\leq p_i\\leq(r-1)/r$, $$ \\prod_{i=1}^r\\mu_{p_i}(\\mathcal{F}_i)\\leq \\prod_{i=1}^r p_i.$$The main dif"},"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":"2607.21589","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-23T17:58:16Z","cross_cats_sorted":["math.CA","math.FA"],"title_canon_sha256":"3848d2a8b2e6c80b08925ccc318a47fd8d21a3298789c3d1be04be611490aebc","abstract_canon_sha256":"53cb1035b7865a605484c847fbb7c1c7c8cebe3ae769afad907f4bb47aad175a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T01:24:41.773973Z","signature_b64":"U/8l6WfgDEgLtHkArEWahsLw8a/Rk24OIOSYkAC8ljWP95S1wsrlqlNQ0AqufjwG1jz+ZrDHeEzx3yMo8yN4Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6640a277e0c025068ca4c5ac0dcb88a13afe4b445ace33b4d6ea9539a6e3c246","last_reissued_at":"2026-07-24T01:24:41.773130Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T01:24:41.773130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Frankl--Tokushige product conjectures for $r$-cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.CO","authors_text":"Fan Chang, Hong Liu, Miao Liu","submitted_at":"2026-07-23T17:58:16Z","abstract_excerpt":"We settle the uniform and biased product conjectures of Frankl and Tokushige for $r$-cross-intersecting families. Let $r\\geq2$, let $0\\leq k_i\\leq(r-1)n/r$, and let $\\mathcal{F}_i\\subseteq\\binom{[n]}{k_i}$ be $r$-cross-intersecting. We prove the sharp inequality $$\\prod_{i=1}^r\\frac{|\\mathcal{F}_i|}{\\binom{n}{k_i}}\\leq \\prod_{i=1}^r\\frac{k_i}{n},$$ with equality attained by the corresponding levels of a common $1$-star. As a consequence, we obtain the analogous $p_i$-biased measure theorem for $0\\leq p_i\\leq(r-1)/r$, $$ \\prod_{i=1}^r\\mu_{p_i}(\\mathcal{F}_i)\\leq \\prod_{i=1}^r p_i.$$The main dif"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21589","kind":"arxiv","version":1},"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/2607.21589/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":"2607.21589","created_at":"2026-07-24T01:24:41.773569+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.21589v1","created_at":"2026-07-24T01:24:41.773569+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.21589","created_at":"2026-07-24T01:24:41.773569+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZAKE57AYASQ","created_at":"2026-07-24T01:24:41.773569+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZAKE57AYASQNDFE","created_at":"2026-07-24T01:24:41.773569+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZAKE57A","created_at":"2026-07-24T01:24:41.773569+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE","json":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE.json","graph_json":"https://pith.science/api/pith-number/MZAKE57AYASQNDFEYWWA3S4IUE/graph.json","events_json":"https://pith.science/api/pith-number/MZAKE57AYASQNDFEYWWA3S4IUE/events.json","paper":"https://pith.science/paper/MZAKE57A"},"agent_actions":{"view_html":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE","download_json":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE.json","view_paper":"https://pith.science/paper/MZAKE57A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.21589&json=true","fetch_graph":"https://pith.science/api/pith-number/MZAKE57AYASQNDFEYWWA3S4IUE/graph.json","fetch_events":"https://pith.science/api/pith-number/MZAKE57AYASQNDFEYWWA3S4IUE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE/action/storage_attestation","attest_author":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE/action/author_attestation","sign_citation":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE/action/citation_signature","submit_replication":"https://pith.science/pith/MZAKE57AYASQNDFEYWWA3S4IUE/action/replication_record"}},"created_at":"2026-07-24T01:24:41.773569+00:00","updated_at":"2026-07-24T01:24:41.773569+00:00"}