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\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}


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