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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 $iMIN(H)$,
\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}
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