无Diamond族的Lubell函数的渐近界
The Asymptotic Bound of the Lubell Function for Diamond-free Families
摘要：
对于$[n]:={1,2,...,n}$的子集族，Lubell函数被定义为$\hb_n(\F):=\sum_{F\in\F}\binom{n}{|F|}^{-1}$。在\cite{GriLiLu}中，Griggs，Lu和作者证明了如果$[n]$的子集的族$\F$不包含形成菱形的四个不同的集合$A$，$B$，$C$和$D$，即$A\subset B\cap C$和$B\cup C\subset D$，则$\hb_n(\F)\le 2+\lfloor\frac{n^2}{4}\rfloor/(n^2-n)$.此外，上界是由三种类型的家庭。本文证明了该猜想的上界是渐近正确的。此外，我们还给出了关于无偏序集族的Lubell函数最大化问题的一些结果。
英文摘要：
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. In this paper, we prove the upper bound in the conjecture is asymptotically correct. In addition, we give some results related to the problem of maximizing the Lubell function for the poset-free families.
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