% AUTOGENERATED FILE; you probably want to edit the .tex.in file, not this.
\documentclass[a4paper]{article}
\pagestyle{headings}
\usepackage{amssymb,amsmath,enumerate,amsthm,url}
\usepackage[all,cmtip]{xy}
\usepackage{graphicx}
\providecommand{\Dashv}{\mathrel{\text{\reflectbox{$\vDash$}}}}
\bibliographystyle{alpha}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{fact}[theorem]{Fact}
\newtheorem*{fact*}{Fact}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem*{claim*}{Claim}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem*{definition*}{Definition}
\newtheorem{notation}[theorem]{Notation}
\newtheorem*{notation*}{Notation}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem*{remark*}{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem*{example*}{Example}
\newtheorem{note}[theorem]{Note}
\newtheorem*{note*}{Note}
\newtheorem{question}[theorem]{Question}
\newtheorem*{question*}{Question}
\begin{document}
An uncountable structure is {\bf quasiminimal} if every definable subset is 
countable or else has countable complement. This paper considers the problem 
of finding quasiminimal models of a given first-order theory, and obtains  
various results under various assumptions.

It was shown in \cite{PT} that the generic type $p_M = \{ \theta(x) : 
|\theta(M)| > \aleph_0 \} \in S_1(M)$ of a well-behaved quasiminimal structure 
$M$ has global heir $p$ which is definable and {\bf strongly regular}: 
each $p\negmedspace\restriction_B$ is orthogonal to any other $q \in S_1(B)$.

This paper conversely considers a first order theory $T$ with a 
definable strongly regular type $p$ and builds quasiminimal models $M$ for 
which $p\negmedspace\restriction_M$ is the generic type. It first observes, using Vaught two-cardinal 
techniques, that such a model of cardinality $\aleph_1$ always exists. Under 
further hypotheses, it finds arbitrarily large models. The first such 
hypothesis is that $T$ admits definable Skolem functions, and $p$ is 
symmetric. The second is that $T$ admits a quasiminimal model of cardinality 
$\geq  \beth_{\omega_1}$ with generic type $p$, since this suffices for an 
Erd\H{o}s-Rado argument; hence the Hanf number for existence of quasiminimal 
models is at most $\beth_{\omega_1}$.

The third is the case that $T$ is stable. The proof here goes via the 
technology of l-isolation and independent systems, building a ``highly 
l-atomic'' model over a Morley sequence in $p$. Finally, the paper considers 
the case of $\omega$-stable $T$ and shows that in this case the class of prime 
models over Morley sequences in $p$ is moreover a ``quasiminimal excellent 
class'', using their characterisation obtained in \cite{BHHKK}.

%\bibliography{constructingQM}
\newcommand{\etalchar}[1]{$^{#1}$}
\begin{thebibliography}{BHH{\etalchar{+}}14}

\bibitem[BHH{\etalchar{+}}14]{BHHKK}
Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kes{\operatorname{a}}l{\operatorname{a}}, and Jonathan
  Kirby.
\newblock Quasiminimal structures and excellence.
\newblock {\em Bull. Lond. Math. Soc.}, 46(1):155--163, 2014.

\bibitem[PT11]{PT}
Anand Pillay and Predrag Tanovi{\'c}.
\newblock Generic stability, regularity, and quasiminimality.
\newblock In {\em Models, logics, and higher-dimensional categories}, volume~53
  of {\em CRM Proc. Lecture Notes}, pages 189--211. Amer. Math. Soc.,
  Providence, RI, 2011.

\end{thebibliography}
\end{document}
