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\begin{document}
\centerline{\bf Homework 05}
\centerline{Morally Due Tue Feb 29 at 3:30PM. Dead Cat March 2 at 3:30PM}
\begin{enumerate}
\item
(0 points)
What is your name? Write it clearly.
When is the take-home midterm due?
\item
(30 points)
Prove the following (the finite 3-hypergraph Ramsey Theorem)
by using the infinite 3-hypergraph Ramsey Theorem.
{\it For all $k,c$ there exists $n$ such that for all $\COL\colon\binom{[n]}{3}\into[c]$
there exists a homog set of size $k$. }
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\item
(30 points)
We will look at the following statement which we call FCR (Finite Can Ramsey)
{\it For all $k,c$ there exists $n$ such that for all $\COL\colon\binom{[n]}{2}\into\omega$
either there is a homog set of size $k$ OR a min-homog set of size $k$ OR a max-homog set of size $k$
OR a rainbow set of size $k$.}
\begin{enumerate}
\item
(10 points) TRY to prove FCR from the infinite Can Ramsey Theorem on Graphs. YOU WILL FAIL.
Where does it fail?
\item
(20 points) Prove FCR somehow (Hint: DO NOT try to finitize Mileti's proof.
That can be done but is messy.)
\end{enumerate}
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\item
(40 points)
In this problem we will look at the following problems:
{\it Given $X\subseteq \R^1$ (or $\S^1$ or $\R^2$) of size $n$ show
there is a large subset of size $\Omega(f(n))$ (you will figure out the $f$)
where all of the distances are different. ($S^1$ is any circle.) }
We will NOT use Can Ramsey.
Let $X$ be a set of points (could be in $\R^1$ or $S^1$ or $\R^2$).
Let $M\subseteq X$. $M$ is {\it d-maximal} if (1) every pair of distances is different and
(2) for all $p\in X-M$, statement (1) is false for $M\cup \{p\}$.
(Note that we DO NOT have the color-degree bound we had in the past.)
\begin{enumerate}
\item
(20 points)
Find an increasing function $f$ such that the following is true:
{\it If $X\subseteq \R^1$ is a set of $n$ points then every d-maximal set is of size
$\Omega(f(n))$.}
\item
(20 points)
Find an increasing function $g$ such that the following is true:
{\it If $X\subseteq \S^1$ is a set of $n$ points then every d-maximal set is of size
$\Omega(g(n))$.}
\end{enumerate}
\end{enumerate}
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