{"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 $A\\in \\mathcal{F}_i,B\\in\\mathcal{F}_j$, $1\\leq i<j\\leq m$, $|A\\bigcap B|\\geq1$. 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,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.18426","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/2411.18426/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"}