{"paper":{"title":"The Asymptotic Bound of the Lubell Function for Diamond-free Families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wei-Tian Li","submitted_at":"2012-06-05T01:31:32Z","abstract_excerpt":"For a family of subsets of $[n]:={1,2,...,n}$, the Lubell function is defined as $\\hb_n(\\F):=\\sum_{F\\in\\F}\\binom{n}{|F|}^{-1}$. In \\cite{GriLiLu}, Griggs, Lu, and the author conjectured that if a family $\\F$ of subset of $[n]$ does not contain four distinct sets $A$, $B$, $C$ and $D$ forming a diamond, namely $A\\subset B\\cap C$ and $B\\cup C\\subset D$, then $\\hb_n(\\F)\\le 2+\\lfloor\\frac{n^2}{4}\\rfloor/(n^2-n)$. Moreover, the upped bound is achieved by three types of families.\n  In this paper, we prove the upper bound in the conjecture is asymptotically correct. In addition, we give some results "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0806","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"}