The Asymptotic Bound of the Lubell Function for Diamond-free Families

TL;DR

This paper proves that the maximum Lubell function for diamond-free families of subsets of [n] asymptotically matches a conjectured bound, advancing understanding of extremal set theory.

Contribution

It confirms the asymptotic correctness of the conjectured upper bound for the Lubell function in diamond-free families.

Findings

The upper bound in the conjecture is asymptotically correct.

Identifies families that achieve the maximum Lubell function.

Provides related results on maximizing Lubell function for poset-free families.

Abstract

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.