{"paper":{"title":"Applications of graph containers in the Boolean lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Zsolt Wagner, Andrew Treglown, Jozsef Balogh","submitted_at":"2016-02-18T16:44:07Z","abstract_excerpt":"We apply the graph container method to prove a number of counting results for the Boolean lattice $\\mathcal P(n)$. In particular, we: (i) Give a partial answer to a question of Sapozhenko estimating the number of $t$ error correcting codes in $\\mathcal P(n)$, and we also give an upper bound on the number of transportation codes; (ii) Provide an alternative proof of Kleitman's theorem on the number of antichains in $\\mathcal P(n)$ and give a two-coloured analogue; (iii) Give an asymptotic formula for the number of $(p,q)$-tilted Sperner families in $\\mathcal P(n)$; (iv) Prove a random version o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05870","kind":"arxiv","version":4},"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"}