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\begin{document}
\centerline{\bf Homework 03}
\centerline{Morally Due Tue Feb 15 at 3:30PM. Dead Cat Feb 17 at 3:30}
\begin{enumerate}
\item
(0 points)
What is your name? Write it clearly.
When is the take-home midterm due?
\item
(35 points)
Give a well written complete proof of Mileti's proof of the 2-ary Can Ramsey Theorem.
(I did the first two steps in class, but you will need to include those as well.)
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\item
(35 points)
In this problem we have a part of a proof, but want a theorem.
Fill in the BLANK and the BLAH BLAH to get a theorem OF INTEREST.
\noindent
{\bf Theorem}
Let $X$ be a countable infinite set of points in the plane, no three colinear.
Then there exists $Y\subseteq X$, $|Y|=\infty$, such that BLANK.
\noindent
{\bf Proof}
Order the points in $X$ arbitrarily, so
$$X = \{ p_1, p_2, p_3, \ldots \}.$$
Define a coloring $\COL\colon\binom{N}{3}\into \R$ via $\COL(i,j,k)$ is the area of
the triangle created by $p_i,p_j,p_k$.
The number of reals used is countable so we can apply Can Ramsey.
Hence there exists $H\subseteq \N$, $|H|=\infty$, $H$ is $A$-homog for some $A\subseteq \{1,2,3\}$.
Look at the set of points
$$Y=\{p_i \colon i\in H\}.$$
Then BLAH BLAH so $Y$ is BLANK.
\noindent
{\bf End of Proof of Theorem}
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\item
(30 points)
{\bf Point of this Problem} The first day of class we proved that no matter
how you color the edges of $K_6$ there will be a monochromatic triangle.
What about $K_5$? It turns out that there IS a coloring of $K_5$ with
NO mono triangles. But how common is that? In this problem you will generate
1000 random 2-colorings of the edges of $K_5$ and COUNT how many have
0 mono triangle, 1 mono triangle, $\ldots$, 10 mono triangles.
You will generate these colorings 9 different ways.
Each time you do it you will count how many of the colorings had
0 mono triangles, 1 mono triangle, $\ldots$, 9 mono triangles.
For $0\le i\le 10$, $n_i$ will be the number that have $i$ mono triangles.
NOTE: All we want to hand in will be the table of data, and some speculation
about theorems, NOT the code itself.
On the next page IS the problem formally.
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{\bf ONE} (0 points but you need to do this for later parts)
Write a program that will take input $0\le p\le 1$ and randomly assign colors
to the edges of $K_5$ with each edge being RED with prob $p$ and BLUE with prob $1-p$.
(You might want to use 0 and 1 instead of RED and BLUE since computers operate that way.)
\smallskip
{\bf TWO} (0 points but you need to do it for later parts)
Write a program that will, given a 2-coloring of $K_5$,
count 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[ ]
For $p= 0.1, 0.2, \ldots, 0.9$
\begin{enumerate}
\item
$n_0=0$, $n_1=0$, $\ldots$, $n_{10}=0$.
(Recall that $n_i$ 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 $K_5$ by coloring RED with prob $p$ and BLUE with prob $1-p$.
(You may want to use colors 0 and 1 instead.)
\item
Find $j$, the number of mono triangles.
\item
$n_j = n_j+1$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
{\bf FOUR} (30 points) Use your program to produce the a table of data
The table should look like what is below except that (1) I made up the numbers, and
(2) your table should not have any DOT DOT DOT in it, it should have all the numbers.
\[
\begin{array}{|c|ccccccccccc|}
\hline
p & n_0 & n_1 & n_2 & n_3 & n_4 & n_5 & n_6 & n_7 & n_8 & n_9 & n_{10} \cr
\hline
0.1 & 0 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 5 & 5 \cr
0.2 & 0 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 7 & 3 \cr
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots& \vdots & \vdots &\vdots & \vdots & \vdots & \vdots \cr
0.9 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr
\hline
\end{array}
\]
\smallskip
{\bf FIVE} (0 points) Looking at the data formulate a conjecture about colorings of $K_5$.
Prove your conjecture.
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\item
(Extra Credit)
Give a well written clean proof of 3-ary Can Ramsey.
There are three ways to do this. The more ways you do,
the more extra credit you get!
\begin{enumerate}
\item
Use some $a$-ary Ramsey Theorem and lots of cases
(with good notation you can consolidate them), and
all cases easy.
\item
Use some $a$-ary Ramsey Theorem with fewer cases than the proof
suggested in Part 1
(with good notation you can consolidate them), and
the rainbow case will need a version of maximal sets.
\item
Use a Mileti-style proof. Note that 2-ary Mileti used
1-ary Can Ramsey. Similarly, 3-ary Mileti will use
2-ary Can Ramsey. It will be similar to the proof
of 3-ary Ramsey from 2-ary Ramsey.
\end{enumerate}
\end{enumerate}
\end{document}