{"paper":{"title":"Set Systems Containing Many Maximal Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Imre Leader, J. Robert Johnson, Paul A. Russell","submitted_at":"2013-09-18T13:47:48Z","abstract_excerpt":"The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power set $\\mathcal{P}(\\{1,2,\\dots,n\\})$? We will show that for each fixed $\\alpha>0$ there is a family of $\\alpha 2^n$ sets containing $(\\alpha+o(1))n!$ such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning bri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4643","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"}