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\begin{document}
\centerline{\bf Homework 2}
\centerline{Morally Due Tue Feb 8 at 3:30PM. Dead Cat Feb 10 at 3:30}
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\begin{enumerate}
\item
(0 points)
What is your name? Write it clearly.
When is the take-home midterm due?
\item
(35 points)
Look at the slides on the Can Ramsey Theorem.
Look at the proof that uses 4-ary Ramsey.
RECAP OF THAT PROOF FOR OUR PURPOSES:
Given a coloring $\COL:\binom{N}{2}\into [\omega]$ we created
$\COL':\binom{N}{4}\into [16]$. We then applied 4-ary Ramsey Theory to get
a homog set $H$ (relative to $\COL'$). We show that whatever color the homog set
was we found an infinite subset of $H$ that was with respect to $\COL$
either (a) homog, (b) max-homog, (c) min-homog, or (d) rainbow.
OKAY, now for our problem.
Look at the case where there is an infinite homog (using coloring $\COL'$) $H$
such that
$$(\forall x_1 < x_2 < x_3 < x_4 \in H)[ \COL(x_2,x_3)=\COL(x_1,x_4)].$$
Show that $H$ (or perhaps some infinite subset of it) is homog with respect to $\COL$.
(I am asking you to do one of the cases I skipped.)
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\item
(35 points) Before the proof of Can Ramsey that used 3-ary Ramsey we proved the following:
{\it Assume $X$ is infinite and $\COL:\binom{X}{2}\into [\omega]$.
Assume that, for all $x\in X$ and colors $c$, $\degg_c(x)\le 1$.
Let $M$ be a MAXIMAL rainbow set.
Then $M$ is infinite.}
In this problem you will come up with and prove a FINITE version of this theorem.
Fill in the function $f(n)$ and prove the following:
{\it Assume $|X|=n$ and $\COL:\binom{X}{2}\into [\omega]$.
Assume that, for all $x\in X$ and colors $c$, $\degg_c(x)\le 1$.
Let $M$ be a MAXIMAL rainbow set.
Then $|M|\ge \Omega(f(n))$.}
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\item
(30 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 countably infinite set of points in the plane.
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 $\COL\colon\binom{N}{2}\into \R$ via $\COL(i,j)=|p_i-p_j|$.
The number of reals used is countable so we can apply Can Ramsey.
Hence there exists $H\subseteq \N$, $|H|=\infty$, $H$ is either homog, min-homog, max-homog,
or rainbow. 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
(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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