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\headers{Math773}{Homework 5}{Due Thurs, April 2, 2015}

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\noindent
\textbf{Directions:} Solve the following open-ended problem, based Frankl's union-closed sets conjecture.

\begin{enumerate}

\item {} Find the largest function $f$ such that the following holds.  If $\F$ is a nonempty, union-closed subset of $2^{[n]}$ with $\emptyset\not\in \F $, then there exists $x\in [n]$ such that $\frac{|\{A\in \F\st x\in A\}|}{|\F|} \ge f(n)$.  Comment: Frankl's conjecture states that $f(n) = 1/2$.  Proving that $f(n) \ge \varepsilon$ for some positive $\varepsilon$ would represent a major advance.  So, it is likely that your function will tend to $0$ as $n\to\infty$.  The goal is to establish the statement for a function $f$ that approaches $0$ as slowly as possible.

\end{enumerate}
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