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\begin{document}

\centerline{\bf Homework 06} 

\centerline{Morally Due Tue March 8 at 3:30PM. Dead Cat March 10 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)
Let $R_a(k)$ be the least $n$ such that 

\centerline{\it for all $\COL\colon\binom{[n]}{a}\into[2]$ there exists a homog set of size $k$.}

For this problem assume $R_2(k)\le 2^{2k}$ (which is true). 

In class I sketched the beginning of the proof that $R_3(k) \le 2^{2^{O(k)}}$. 

For this problem give a complete rigorous proof. 

\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(35 points)
Prove the following:

{\it For all $k$ 
there exists $n$ such that for all $COL:\binom{\{k,\ldots,n\}}{1}\into \omega$ 
there exists either 
\begin{itemize}
\item
a LARGE homog set, or 
\item
a LARGE rainbow set (all the numbers are colored differently).
\end{itemize}
}

\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(30 points) 
Prove the following:
{\it For all $k$ 
there exists $n$ such that for all $COL:\binom{\{k,\ldots,n\}}{2}\into [100]$ 
there exists an $H\subseteq [n]$ such that 
\begin{itemize}
\item
$H$ is a homog set, and
\item
$|H|\ge 2^{2^{\min(H)}}$.
\end{itemize}
}



\end{enumerate}

\end{document}