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The families $\\mathcal{F}_1\\subseteq \\binom{[n]}{R_1},\\mathcal{F}_2\\subseteq \\binom{[n]}{R_2},\\ldots,\\mathcal{F}_m\\subseteq \\binom{[n]}{R_m}$ are said to be non-empty cross-intersecting if for each $i\\in [m]$, $\\mathcal{F}_i\\neq\\emptyset$ and for any $A\\in \\mathcal{F}_i,B\\in\\mathcal{F}_j$, $1\\leq i<j\\leq m$, $|A\\bigcap B|\\geq1$. In this paper, we determine the maximum value of $\\sum_{j=1}^{m}|\\mathcal{F}_j|$ for non-empty cross-intersecting family $\\mathcal{F}_1,"},"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":"2411.18426","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-27T15:08:45Z","cross_cats_sorted":[],"title_canon_sha256":"fec663a6b5b828ce8ea7e326774e216f0d3ac0d88bd0cad4f0eef61cada02ceb","abstract_canon_sha256":"6edc2ed20cf14a020def26e9d6048f4f715864e8eafa509c8054996b05315bb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:41:16.449809Z","signature_b64":"c0FGEZIcDGRB+dJOYBCrdq3M4i/SrKN4iyc0C0+BqSZxfAepKUdDr7MHL1jJ7vLjKnmEC45CGxQCpLD6xUljBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5722a64b1fcde0a26f15c136718466ddd49644f42142456f5339edbedc6204b7","last_reissued_at":"2026-07-05T09:41:16.449330Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:41:16.449330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-uniform Cross-intersecting Families","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huajun Zhang, Jimeng Xiao, Qing Xiang, Zhen Jia","submitted_at":"2024-11-27T15:08:45Z","abstract_excerpt":"Let $m\\geq 2$, $n$ be positive integers, and $R_i=\\{k_{i,1} >k_{i,2} >\\cdots> k_{i,t_i}\\}$ be subsets of $[n]$ for $i=1,2,\\ldots,m$. The families $\\mathcal{F}_1\\subseteq \\binom{[n]}{R_1},\\mathcal{F}_2\\subseteq \\binom{[n]}{R_2},\\ldots,\\mathcal{F}_m\\subseteq \\binom{[n]}{R_m}$ are said to be non-empty cross-intersecting if for each $i\\in [m]$, $\\mathcal{F}_i\\neq\\emptyset$ and for any $A\\in \\mathcal{F}_i,B\\in\\mathcal{F}_j$, $1\\leq i<j\\leq m$, $|A\\bigcap B|\\geq1$. 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