{"paper":{"title":"Families with no $s$ pairwise disjoint sets","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":"2016-07-20T20:55:08Z","abstract_excerpt":"For integers $n\\ge s\\ge 2$ let $e(n,s)$ denote the maximum of $|\\mathcal F|,$ where $\\mathcal F$ is a family of subsets of an $n$-element set and $\\mathcal F$ contains no $s$ pairwise disjoint members. Half a century ago, solving a conjecture of Erd\\H os, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all $m,s\\ge 1$. During the years very little progress in the general case was made.\n  In the present paper we state a general conjecture concerning the value of $e(sm-l,m)$ for $1<l<s$ and prove its validity for $s>s_0(l,m).$ For $l=2$ we determine the value of $e(sm-2,m)$ for all $s\\ge 5.$\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06122","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"}