{"paper":{"title":"Kleitman's conjecture about families of given size minimizing the number of $k$-chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Zsolt Wagner, Jozsef Balogh","submitted_at":"2016-09-08T04:20:46Z","abstract_excerpt":"A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family $\\mathcal{F}\\subseteq \\mathcal{P}(n)$ that does not contain a $2$-chain $F_1\\subsetneq F_2$. Erd\\H{o}s later extended this result and determined the largest family not containing a $k$-chain $F_1\\subsetneq \\ldots \\subsetneq F_k$. Erd\\H{o}s and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result.\n  This question was resolved for $2$-chains by Kleitman in $1966$, who showed that amongst families of size $M$ in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02262","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"}