{"paper":{"title":"A sharp product bound for non-trivial cross-intersecting families","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yang Huang","submitted_at":"2026-06-22T13:33:17Z","abstract_excerpt":"Two families $\\mathcal{A}, \\mathcal{B} \\subset \\binom{[n]}{k}$ are cross-intersecting if $A \\cap B \\ne \\emptyset$ for all $A \\in \\mathcal{A}$ and $B \\in \\mathcal{B}$, and non-trivial if neither $\\m A$ nor $\\m B$ is a star. Pyber proved that any two cross-intersecting families $\\mathcal{A}, \\mathcal{B} \\subset \\binom{[n]}{k}$ satisfy $|\\mathcal{A}||\\mathcal{B}| \\le \\binom{n-1}{k-1}^2$, and the maximum is attained by two full stars. Frankl, as well as Frankl and Wang, conjectured that the sharp bound, when both families are required to be non-trivial, is $h(n,k)^2$, where $h(n,k) = \\binom{n-1}{k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23322","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/2606.23322/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"}