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\begin{document}

\centerline{\bf Homework 07} 

\centerline{Morally Due Tue March 29 at 3:30PM. Dead Cat March 31 at 3:30}

{\bf IN THIS HW WHENEVER I SAY ``A SET OF POINTS IN THE PLANE'' I MEAN THAT
THEY HAVE NO THREE COLINEAR.}

\begin{enumerate}

\item
(0 points)
What is your name?  Write it clearly.
When is the take-home final due?

\item
(35 points)
Let $N(k)$ be the least $n$ such that for all sets of $n$ points there is a subset of
$k$ of them that form a convex $k$-gon. 

We begin a proof that $N(k)$ exists and you need to finish it.

{\it We show that $n=R_3(k)$ suffice. Let $X$ be a set of $n=R_3(k)$ points in
the plane. Let the points be $p_1,p_2,\ldots,p_n$.

Color $(p_i,p_j,p_j)$ (with $i<j<k$) RED if $p_i,p_j,p_k$ is clockwise.

Color $(p_i,p_j,p_j)$ (with $i<j<k$) BLUE if $p_i,p_j,p_k$ is counter clockwise.

The Homogenous set of size $k$ form a convex $k$-gon because FILL THIS IN.}



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\item
(35 points)

{\bf Def} $\JULY(k)$ is the least $n\ge$ such that
for all 2-colorings of $\binom{\{k,\ldots,n\}}{2}$ there exists
a set $H$ such that
\begin{itemize}
\item
$|H|\ge 3$,
\item
$|H|>MIN(H)$,
\item 
$\COL$ restricted to $\binom{H}{2}$ is constant.
\end{itemize}

Find a number $A$ such that you can prove $\JULY(1)\le A$. 

(I have a proof with $A=8$ but given that the original version of
this problem was incorrect, I am phrasing it this way so it can't 
go wrong.)

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\item
(30 points) 
Recall:

{\it If $n\equiv 1 \pmod 2$ then for any $\COL\colon\binom{[n]}{2}\into[2]$
there exists at least 

$$\frac{n^3}{24} -\frac{n^2}{4} + \frac{5n}{24}$$

monochromatic $K_3$'s.} 

We will vary this in two ways.

\begin{enumerate}
\item 
(15 points) 
Find a function $f$ such that the followings is true:

{\it If $n\equiv 0 \pmod 2$ then for any $\COL\colon\binom{[n]}{2}\into[2]$
there exists at least $f(n)$ monochromatic $K_3$'s.} 

Prove your result. 

\item 
(15 points) 
We are interested in what happens if you have THREE colors.
Do some empirical studies to try to find a
function $f$ such that the following holds:

{\it If $\COL\colon\binom{[n]}{2}\into[3]$ then 
there exists at least $f(n)$ monochromatic $K_3$'s.
($f(n)$ an be approximate. For example, if the problem was for 2-coloring
then $f(n)$ could be $\frac{n^3}{24}$.) }

(HINT: Use the code you wrote for the midterm; however, only use the case of
$p_1=p_2=p_3=\frac{1}{3}$.) 

\item
(Extra Credit, 0 points) PROVE a result along the lines of:

{\it If $n$ satisfies condition YOU FILL IN and $\COL\colon\binom{[n]}{2}\into[3]$ then 
there exists at least $f(n)$ monochromatic $K_3$'s.)

(HINT: Use the code you wrote for the midterm; however, only use the case of
$p_1=p_2=p_3=\frac{1}{3}$.) }

\end{enumerate}

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\item
(Extra Credit, but THINK ABOUT IT. WARNING- I have not done this problem)

{\it Let $X$ be an infinite set of points $p_1,p_2,p_3,\ldots$. Let $\COL\binom{\N}{3}\into \omega$
be defined as follows:

$$\COL(i,j,k) = \hbox{ the number of points inside the $(i,j,k)$ triangle.}$$

Apply the 3-ary Can Ramsey Theorem to this Coloring. NOW WHAT?

}

\item
(Extra Credit, but THINK ABOUT IT--WARNING: the way I know how to do this is based on
material you have not seen) We want to write a sentence $\phi$ in the language of graphs
such that

$G\models \phi$ IFF $G$ has an even number of vertices.

Is this possible? 


\end{enumerate}

\end{document}