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\begin{document}

$\;$\hfill HW12 Due 2012/12/12 12:12

\begin{center}
{\LARGE\bf Homework 12}
\end{center}

\begin{hw} Let $n$ be a positive integer and $\cf$ be an intersecting family in $2^{[n]}$ and $|\cf| = 2^{n-1}$. Prove that
$\cf$ is a filter. (i.e., $B \in \cf$ whenever there is some $A \in \cf$ and $A \subseteq B$.)
\end{hw}

\begin{hw} Describe a family $\cf$ of $120$ sets in $2^{[12]}$, where each set is of size 5,
and any two sets $A, B \in \cf$ have $|A \cap B| \geq 2$.
\end{hw}

\begin{hw} Describe a family $\cf$ of {\em more than}  $30$ sets in $2^{[11]}$, where each set is of size 5,
and any two sets $A, B \in \cf$ have $|A \cap B| \geq 3$.
\end{hw}

\begin{hw} Given $n$ real numbers $a_1$, ..., $a_n$ such that each $|a_i| \geq 1$. For any subset $A \subseteq [n]$,
define $S(A) = \sum_{i \in A} a_i$. If $A_1$, ..., $A_m$ are $m$ distinct subsets of $[n]$ such that $|S(A_i) - S(A_j)| < 1$ for any $i \neq j$, how big can $m$ be?
Justify your answer. (i.e., For each $n$, find a number $f(n)$, prove that that we always have $m \leq f(n)$, and show an example
where $m = f(n)$.)
\end{hw}

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