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{\bf CS 174: Combinatorics and Discrete Probability} \hfill Fall 2012 \bigskip \\
\centering{\Large Homework 7} \medskip \\
\centering{Due: Thursday, October 25, 2012 by {\bf 9:30am}}
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} \bigskip \\
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\noindent{\it \textbf{Instructions}}: {\it You should upload your homework
solutions on bspace. You are strongly encouraged to type out your solutions
using \LaTeX . You may also want to consider using mathematical mode typing in
some office suite if you are not familiar with \LaTeX . If you must handwrite
your homeworks, please write clearly and legibly. We will not grade homeworks
that are unreadable. You are encouraged to work in groups of 2-4, but you {\bf
must} write solutions on your own. Please review the homework policy carefully
on the class homepage.} \medskip \\
\noindent {\bf Note}: You \emph{must} justify all your answers. In particular, you will get
no credit if you simply write the final answer without any explanation. \medskip
\\
\noindent {\bf Problem 1}. {\it (Exercise 6.10 from MU -- 6 points)} A family of
subsets ${\mathcal F}$ of $\{1, 2, \ldots, n\}$ is called an \emph{antichain} if
there is no pair of sets $A$ and $B$ in ${\mathcal F}$ satisfying $A \subset B$.
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\begin{enumerate}
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\item[(a)] Given an example of ${\mathcal F}$ where $|{\mathcal F}| = {n \choose
\lfloor n/2 \rfloor}$.
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\item[(b)] Let $f_k$ be the number of sets in ${\mathcal F}$ with size $k$. Show
that
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\[ \sum_{k=0}^n \frac{f_k}{{n \choose k}} \leq 1.\]
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({\it Hint}: Choose a random permutation of the numbers from $1$ to $n$, and let
$X_k = 1$ if the first $k$ numbers in your permutation yeild a set in ${\mathcal
F}$. If $X = \sum_{k=0}^n X_k$, what can you say about $X$?)
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\item[(c)] Argue that $|{\mathcal F}| \leq {n \choose \lfloor n/2 \rfloor}$ for
any antichain ${\mathcal F}$.
\end{enumerate} \medskip
\noindent {\bf Problem 2}. {\it (Exercise 6.14 from MU -- 6 points)} Consider a
graph in $G_{n, p}$, with $p = 1/n$. Let $X$ be the number of triangles in the
graph, where a trianle is a clique with three edges. Show that
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\[ Pr(X \geq 1) \leq 1/6 \]
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and that
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\[ \lim_{n \rightarrow \infty} \Pr(X \geq 1) \geq 1/7 \]
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({\it Hint}: Use the conditional expectation inequality.) \medskip \\
\noindent {\bf Problem 3} {\it (Exercise 6.18 from MU -- 6 points)}
Let $G = (V, E)$ be an undirected graph and suppose each $v \in V$ is associated
with a set $S(v)$ of $8r$ colours, where $r \geq 1$. Suppose, in addition, that
for each $v \in V$ and $c \in S(v)$ there are at most $r$ neighbours $u$ of $v$
such that $c$ lies in $S(u)$. Prove that there is a proper colouring of $G$
assigning to each vertex $v$ a colour from its class $S(v)$ such that, for any
edge $(u, v) \in E$, the colours assigned to $u$ and $v$ are different. You may
want to let $A_{u, v, c}$ be the event that $u$ and $v$ are both coloured with
colour $c$ and then consider the family of such events. \medskip \\
\noindent {\bf Problem 4} {\it (12 points)} In this problem we will see that the
value $p = \ln(n)/n$ is a \emph{threshold} property that a random graph in the
$G_{n, p}$ model has an isolated vertex, \ie a vertex with no adjacent edges.
That is, we will prove that
\[
\lim_{n \rightarrow \infty} \Pr[ G \mbox{ has an isolated vertex} ] =
\begin{cases} 0 & \mbox{ if } p = \omega(\frac{\ln(n)}{n}) \\
1 & \mbox{ if } p = o(\frac{\ln(n)}{n}) \end{cases}.
\]
\begin{enumerate}
\item[(a)] Let $X$ be the random variable denoting the number of isolated
vertices in $G$. Write down the expectation of $X$ as a function of $n$ and $p$.
\item[(b)] Show that $\E[X] \rightarrow 0$ for $p = \omega(\frac{\ln(n)}{n})$,
and that $\E[X] \rightarrow \infty$ for $p = o(\frac{\ln(n)}{n})$.
\item[(c)] Deduce from part (b) that $\Pr[G \mbox{ has an isolated vertex}]
\rightarrow 0$ for $p = \omega(\ln(n)/n)$.
\item[(d)] Compute $\var(X)$ as a function of $n$ and $p$.
\item[(e)] Deduce from parts (b) and (d) that $\Pr[G \mbox{ has an isolated
vertex}] \rightarrow 1$ for $p = o(\ln(n)/n)$.
\end{enumerate}
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