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\begin{center}{\huge  Nonisolated Regular Types in Superstable
Theories}\end{center}
\begin{center}{\large Predrag Tanovic\\Matematicki Institut SANU\\Knez
Mihajlova
35, Beograd}\end{center}

\bigskip
\large

Let $T$ be a countable, superstable theory and ${\cal M}$ its monster model.
\bigskip

\noindent{\bf Definition}  Let $p \in S(A)$ be nonalgebraic.

\begin{enumerate}
\item[{(a)}]  $p$ is strongly nonisolated if for all finite $B$ and all
formulas $\varphi(\overline x)$ over $AB$
consistent  with $T$, there exists $q \in [\varphi]_{AB}$ such that $p \bot^aq$.
\smallskip

\item[{(b)}]  $p$ is eventually strongly nonisolated, or ${\rm esn}$ for
short, if there is a finite set $B$ and
$a$ nonforking extension $q \in S(AB)$ which is strongly nonisolated.
\end{enumerate}

We discuss the following:
\medskip

\noindent{\bf Theorem}  Suppose $p \in S(\emptyset)$ is regular, stationary
and has limit ordinal $U$-rank.   The
following are all equivalent:

\begin{enumerate}
\item[{(I)}]  $p$ is eventually strongly nonisolated;
\smallskip

\item[{(II)}]  There is an integer $n$ such that for no formula
$\varphi(x_1,x_2,\dots,x_n)$ we have:
\smallskip
$$
(\varphi(x_1,x_2,\dots,x_n) \wedge p(x_1) \wedge p(x_2)\wedge\dots\wedge
p(x_n)) \Leftrightarrow
p^n(x_1,x_2,\dots,x_n)\,;
$$

\item[{(III)}]  There exists a finite set $A$ and a sequence $\{q_n\mid
n\odot \omega\}$ of types extending $p$
to $S(A)$ such that
$$
bnd(q_0) < bnd(q_1) <\, \dots \,< bnd(q_n) < bnd(q_{n+1})
<\,\dots\quad\mbox{and}\quad
\lim_{n\to\infty}bnd(q_n) = bnd(p)\,.
$$
\end{enumerate}
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