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\begin{document}

\begin{center}
\begin{Large}
{\bf Math 2001 - Assignment 12}
\end{Large}
\bigskip

Due April 15, 2026
\end{center}

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\bigskip

\begin{enumerate}
\item \cite[Ex 2.5.28c,d]{Ma:MT}
  Generalize the example from class to show that there exist complete theories
  $T_n$ with exactly $n\geq 4$ countable models. 

\item \cite[Ex 5.2.2]{TZ:CMT}
  Show that the theory of an equivalence relation with two infinite classes has quantiﬁer elimination and is
  $\omega$-stable. Is it $\aleph_1$-categorical?
  
\item 
 Use the Compactness Theorem to show that the infinite version of Ramsey’s Theorem implies the finite version.

\item
  Show that every infinite partially ordered set $(A,<)$ contains an infinite chain (linearly ordered subset)
  or antichain (subset of incomparable elements).

  \smallskip
  
 \noindent Hint: Ramsey's Theorem. 
 
\end{enumerate}


\begin{thebibliography}{1}
%\bibitem{En:MIL}
% Enderton. A Mathematical Introduction to Logic. Harcourt/Academic
%Press, 2nd edition, 2001. 

\bibitem{Ma:MT}
 Marker. Model Theory: An Introduction. Springer, 2002.

\bibitem{TZ:CMT}
 Tent, Ziegler. A Course in Model Theory. Lecture Notes in Logic, Cambridge, 2012. 
\end{thebibliography}

\end{document}





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