{"paper":{"title":"Diamond-free Families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jerrold R. Griggs, Linyuan Lu, Wei-Tian Li","submitted_at":"2010-10-26T04:16:38Z","abstract_excerpt":"Given a finite poset P, we consider the largest size La(n,P) of a family of subsets of $[n]:=\\{1,...,n\\}$ that contains no subposet P. This problem has been studied intensively in recent years, and it is conjectured that $\\pi(P):= \\lim_{n\\rightarrow\\infty} La(n,P)/{n choose n/2}$ exists for general posets P, and, moreover, it is an integer. For $k\\ge2$ let $\\D_k$ denote the $k$-diamond poset $\\{A< B_1,...,B_k < C\\}$. We study the average number of times a random full chain meets a $P$-free family, called the Lubell function, and use it for $P=\\D_k$ to determine $\\pi(\\D_k)$ for infinitely many "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5311","kind":"arxiv","version":3},"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"}