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\begin{document}

\centerline{\bf Practice Problems for Midterm}
\centerline{\bf Not to be handed in}

\newif{\ifshowsoln}

\begin{enumerate}

\item
For every theorem $T$ in this course which had

{\it for all $\COL \colon BLAH \into [2]$ $\ldots$}

consider the following

\begin{enumerate}
\item
Is the theorem still true if I replace 2 by 3?
If so then prove it. 
If not then give a counterexample.
If not then see if some modification works.
\item
Is the theorem still true if I replace 2 by $\omega$. 
If so then prove it. 
If not then give a counterexample.
If not then see if some modification works.
\end{enumerate}

\newpage 

\item
Let $(X_1,\preceq_1)$, $\cdots$, $(X_n,\preceq_n)$ are wqo

Consider the order 

$(X_1\times \cdots \times X_n,\preceq)$

where

$(x_1,\ldots,x_n) \preceq (y_1,\ldots,y_n)$ iff $(\forall i)[x_i\preceq_i y_i]$.

Show that this is a wqo. 

\newpage

\item
Find $c$ such that the following is true. 

{\it For all $\COL \colon \binom{\omega + \Z}{2}\into [10^{100}]$ there 
exists a $c$-homog set that is order equivalent to $\omega+\Z$. }

(Also try to find a $c$ that is optimal.) 

\end{enumerate}

\end{document}

