\documentclass[12pt]{article}
\usepackage{amsmath,mathrsfs,bbm}
\usepackage{amssymb}
\textwidth=4.825in
\overfullrule=0pt
\thispagestyle{empty}

\begin{document}
\noindent
%
%
{\bf Victor Falgas-Ravry}
%
%

\medskip
\noindent
%
%
{\bf Minimal Weight in Union-Closed Families}
%
%

\vskip 5mm
\noindent
%
%
%
%
Let $\Omega$ be a finite set and let $\mathcal{S} \subseteq
\mathcal{P}(\Omega)$ be a set system on $\Omega$. For $x\in \Omega$,
we denote by $d_{\mathcal{S}}(x)$ the number of members of
$\mathcal{S}$ containing $x$. A long-standing conjecture of Frankl
states that if $\mathcal{S}$ is union-closed then there is some $x\in
\Omega$ with $d_{\mathcal{S}}(x)\geq \frac{1}{2}|\mathcal{S}|$.

We consider a related question. Define the \emph{weight} of a family
$\mathcal{S}$ to be $w(\mathcal{S}) := \sum_{A \in \mathcal{S}}
|A|$. Suppose $\mathcal{S}$ is union-closed. How small can
$w(\mathcal{S})$ be? Reimer showed \[w(\mathcal{S}) \geq \frac{1}{2}
|\mathcal{S}| \log_2 |\mathcal{S}|,\] and that this inequality is
tight.  In this paper we show how Reimer's bound may be improved if we
have some additional information about the domain $\Omega$ of
$\mathcal{S}$: if $\mathcal{S}$ separates the points of its domain,
then \[w(\mathcal{S})\geq \binom{|\Omega|}{2}.\] This is stronger than
Reimer's Theorem when $\vert \Omega \vert > \sqrt{|\mathcal{S}|\log_2
|\mathcal{S}|}$. In addition we construct a family of examples showing
the combined bound on $w(\mathcal{S})$ is tight except in the region
$|\Omega|=\Theta (\sqrt{|\mathcal{S}|\log_2 |\mathcal{S}|})$, where it
may be off by a multiplicative factor of $2$.

Our proof also gives a lower bound on the average degree: if
$\mathcal{S}$ is a point-separating union-closed family on $\Omega$,
then \[ \frac{1}{|\Omega|} \sum_{x \in \Omega} d_{\mathcal{S}}(x) \geq
\frac{1}{2} \sqrt{|\mathcal{S}| \log_2 |\mathcal{S}|}+ O(1),\] and
this is best possible except for a multiplicative factor of $2$.

\end{document}

.
