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\begin{document}
\centerline{\bf Take Home Midterm}
\centerline{Morally Due Tue March 15 at 3:30PM. Dead Cat March 17 at 3:30}
\begin{enumerate}
\item
(0 points)
What is your name? Write it clearly.
\item
(25 points)
Prove the following and fill in the $f(k)$.
{\it Theorem} For all $k$ there exists $n=f(k)$ such that the following holds.
For all pairs of colorings:
$\COL_1\colon \binom{[n]}{1} \into [2]$,
$\COL_2\colon \binom{[n]}{2} \into [2]$
there exists $H\subseteq [n]$ and colors $c_1,c_2\in \{1,2\}$ (it's okay if $c_1=c_2$)
such that
\begin{itemize}
\item
$H$ is of size $k$,
\item
every element of $H$ is colored $c_1$, and
\item
every element of $\binom{H}{2}$ is colored $c_2$.
\end{itemize}
\vfill
\centerline{\bf GO TO NEXT PAGE}
\newpage
\item
(25 points)
Let $T$ be the set of trees and $\preceq$ be the minor ordering.
Show that $(T,\preceq)$ is a wqo.
You may use any theorem that was PROVEN in class or on the HW.
(Note that we DID NOT prove the Graph Minor Theorem, so you can't use that.)
\vfill
\centerline{\bf GO TO NEXT PAGE}
\newpage
\item
(25 points)
Let $\Q$ be the rationals.
PROVE or DISPROVE:
{\it For every $\COL\colon \Q \into [100]$
there exists an $H\subseteq \Q$ and a color $c$ such that
\begin{itemize}
\item
$H$ has the same order type as the rationals (so $H$ is a countable set without endpoints
where between any two elements is an element), and
\item
every number in $H$ is the same color.
\end{itemize}
}
\vfill
\centerline{\bf GO TO NEXT PAGE}
\newpage
\item
(25 points)
{\bf ONE} (0 points but you need to do this for later parts)
Write a program such that:
\begin{itemize}
\item
{\it Input} is $n\in\N$ and $0\le p_1,p_2,p_3\le 1$ with $p_1+p_2+p_3=1$ and $p_1\le p_2\le p_3$.
\item
{\it Output} is a $\COL\colon\binom{[n]}{2}\into [3]$ that is generated randomly
with each edge being colored 1 with prob $p_1$, 2 with prob $p_2$, and 3 with prob $p_3$.
\end{itemize}
\smallskip
{\bf TWO} (0 points but you need to do it for later parts)
Write a program that will, given $\COL\colon\binom{[n]}{2}\into [3]$,
counts how many monochromatic triangles it has.
{\bf THREE} (0 points but you need this for later parts)
Write a program that does the following (I use psuedocode.)
\begin{enumerate}
\item
Input $n$. ($n$ will be $\ge 6$.)
\item[ ]
For $p_1,p_2,p_3\in \{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8\}$ such that
$p_1+p_2+p_3=1$ and $p_1\le p_2\le p_2$.
\begin{enumerate}
\item[ ]
$L=\binom{n}{3}$ (the number of triangles in $K_n$).
$n_0=0$, $n_1=0$, $\ldots$, $n_{L}=0$.
($n_i$ will be the number of colorings that have $i$ mono triangles.
Initially this is 0.)
\item
For $i=1$ to 1000
\begin{enumerate}
\item[ ]
Randomly color the edges of $K_n$ by coloring 1 with prob $p_1$, 2 with prob $p_2$, 3 with prob $p_3$.
\item
Find $j$, the number of mono triangles.
\item
$n_j = n_j+1$.
\end{enumerate}
\item
$n_{\max} = \max\{n_0,\ldots,n_L\}$.
\item
$j_{\max}$ is the $j$ such that $n_j=n_{\max}$. (If there is more than one $j$, which is unlikely, take the
least one.)
\end{enumerate}
\end{enumerate}
{\bf FOUR} (25 points) Use your program to produce tables of data.
Our interest is in which $n_j$'s are always 0 and which $n_j$ occurs the most often.
The tables should look like what is below except that I made up the answers.
$n=5$
\[
\begin{array}{|ccc|ccc|}
\hline
p_1 & p_2 & p_3 & \{ j \colon n_j=0 \} & j_{\max} & n_{\max} \cr
\hline
0.1 & 0.1 & 0.8 & \{3,8\} & 3 & 109\cr
0.1 & 0.2 & 0.7 & \{1,9\} & 7 & 108\cr
0.1 & 0.3 & 0.6 & \{2,4,8\} & 1 & 200\cr
0.1 & 0.4 & 0.5 & \{2\} & 1 & 10 \cr
\hline
0.2 & 0.2 & 0.6 & \{1,2,4\} & 3 & 300\cr
0.2 & 0.3 & 0.5 & \{3,4,5,7\} & 2 & 401\cr
0.2 & 0.4 & 0.4 & \{1,2,9\} & 10 & 512\cr
\hline
0.3 & 0.3 & 0.4 & \{2,3,8\} & 7 & 70\cr
\hline
\end{array}
\]
$n=6$
SIMILAR TO ABOVE
$\vdots$
$n=10$
SIMILAR TO ABOVE
{\bf FIVE} (0 points) Looking at the data formulate a conjecture about colorings of $K_n$.
{\bf Extra Credit} Prove your conjecture.
\end{enumerate}
\end{document}