{"paper":{"title":"A size-sensitive inequality for cross-intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2016-03-03T00:17:28Z","abstract_excerpt":"Two families $\\mathcal A$ and $\\mathcal B$ of $k$-subsets of an $n$-set are called cross-intersecting if $A\\cap B\\ne\\emptyset$ for all $A\\in \\mathcal A, B\\in \\mathcal B $. Strengthening the classical Erd\\H os-Ko-Rado theorem, Pyber proved that $|\\mathcal A||\\mathcal B|\\le {n-1\\choose k-1}^2$ holds for $n\\ge 2k$. In the present paper we sharpen this inequality. We prove that assuming $|\\mathcal B|\\ge {n-1\\choose k-1}+{n-i\\choose k-i+1}$ for some $3\\le i\\le k+1$ the stronger inequality $$|\\mathcal A||\\mathcal B|\\le \\Bigl({n-1\\choose k-1}+{n-i\\choose k-i+1}\\Bigr)\\Bigl({n-1\\choose k-1}-{n-i\\choose"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00936","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":""},"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"}