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 $p_M = \{ \theta(x) : 
|\theta(M)| > \aleph_0 \} (- S_1(M)$ of a well-behaved quasiminimal structure 
$M$ has global heir $p$ which is definable and {\bf strongly regular}: 
each $p|_B$ is orthogonal to any other $q (- S_1(B)$.

This paper conversely considers a first order theory $T$ with a 
definable strongly regular type $p$ and builds quasiminimal models $M$ for 
which $p|_M$ 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 
$>= \beth_{\omega_1}$ 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{\"a}l{\"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}
