{"paper":{"title":"On the number of containments in $P$-free families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Bal\\'azs Patk\\'os, D\\'aniel Gerbner, D\\'aniel T. Nagy, M\\'at\\'e Vizer","submitted_at":"2018-04-04T21:06:53Z","abstract_excerpt":"A subfamily $\\{F_1,F_2,\\dots,F_{|P|}\\}\\subseteq \\mathcal F$ is a copy of the poset $P$ if there exists a bijection $i:P\\rightarrow \\{F_1,F_2,\\dots,F_{|P|}\\}$ such that $p\\le_P q$ implies $i(p)\\subseteq i(q)$. A family $\\mathcal F$ is $P$-free, if it does not contain a copy of $P$. In this paper we establish basic results on the maximum possible number of $k$-chains in a $P$-free family $\\mathcal F\\subseteq 2^{[n]}$. We prove that if the height of $P$, $h(P) > k$, then this number is of the order $\\Theta(\\prod_{i=1}^{k+1}\\binom{l_{i-1}}{l_i})$, where $l_0=n$ and $l_1\\ge l_2\\ge \\dots \\ge l_{k+1}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01606","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":""},"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"}