{"paper":{"title":"Two problems on matchings in set families - in the footsteps of Erd\\H{o}s and Kleitman","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":"2016-07-20T21:05:50Z","abstract_excerpt":"The families $\\mathcal F_1,\\ldots, \\mathcal F_s\\subset 2^{[n]}$ are called $q$-dependent if there are no pairwise disjoint $F_1\\in \\mathcal F_1,\\ldots, F_s\\in\\mathcal F_s$ satisfying $|F_1\\cup\\ldots\\cup F_s|\\le q.$ We determine $\\max |\\mathcal F_1|+\\ldots +|\\mathcal F_s| $ for all values $n\\ge q,s\\ge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman.\n  The well-known Erd\\H os Matching Conjecture suggests the largest size of a family $\\mathcal F\\subset {[n]\\choose k}$ with no $s$ pairwise disjoint sets. After more than 50 years its full solution "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06126","kind":"arxiv","version":3},"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"}