{"paper":{"title":"Exact extremal non-trivial cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Biao Wu, Huajun Zhang","submitted_at":"2024-11-25T04:56:03Z","abstract_excerpt":"Two families $\\mathcal{A}$ and $\\mathcal{B}$ of sets are called cross-intersecting if each pair of sets $A\\in \\mathcal{A}$ and $B\\in \\mathcal{B}$ has nonempty intersection. Let $\\cal{A}$ and ${\\cal B}$ be two cross-intersecting families of $k$-subsets and $\\ell$-subsets of $[n]$. Matsumoto and Tokushige [J. Combin. Theory Ser. A 52 (1989) 90--97] studied the extremal problem of the size $|\\cal{A}||\\cal{B}|$ and obtained the uniqueness of extremal families whenever $n\\ge 2 \\ell\\ge 2k$, building on the work of Pyber. This paper will explore the second extremal size of $|\\cal{A}||\\cal{B}|$ and ob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.16091","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/2411.16091/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"}