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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.

Author

Li, Wei-Tian

Published

2012

Language

EN