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\begin{document}
\title{Math 301: Homework 9}
\author{Mary Radcliffe}
\date{due 23 Nov 2015}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Prove Dilworth's Theorem using Hall's Theorem.

\item Complete problems 9 \& 14 from \href{http://people.math.gatech.edu/~trotter/book.pdf}{Applied Combinatorics} on page 125 (figure 6.18 is on page 126).

\item Let $A$ be a set of natural numbers, having size $n$. Prove that there exists a subset $S\subset A$ having $|S|\geq \sqrt{n}$ such that one of the two properties below is true.
\begin{itemize}
\item For every distinct $x, y\in S$, either $x|y$ or $y|x$.
\item Every pair of distinct $x, y\in S$ are mutually indivisible.
\end{itemize}


\end{enumerate}

\end{document}