Let $T$ be an stable theory. 
Further we work in the monster model of $T^{eq}$.
We say that a chain of types of the form
$$tp(a\_1/A\_1)\subset tp(a\_2/A\_2) ... \subset tp(a\_n/A\_n)$$
is a *forking chain* if for every $1< i \le n$
the type $tp(a\_i/A\_i)$ forks over $A\_{i-1}$. 


What can we say about the length of forking chains? 


For example in strongly minimal theories such a chain (in the home sort) has an maximum length of 1. Moreover, there exists no chain of length $|T|^+$ if and only if $T$ is simple. This is since forking has local character (every type does not fork over a set of size $<|T|$) if and only if $T$ is simple.
For theory of Morley rank $N$ such a chain is bounded by $N$. But what about the lower bounds? 


Can we find a chain of length $n$ in a theory with Morley rank $\ge n$? 


Does any type $p$ of Morley rank $n$ start an forking chain of length $n$?


Do theories without Morley rank have chains of length $|T|$? 



Chains of forking extension in stable theories
The answer is no to all three questions.


First notice that if there exists a maximal finite forking chain, 
 then its length is the SU-rank 
 (or U-rank in the stable context) of $tp(a\_1/A\_1)$. 
Since there are theories of Morley-rank $>1$ and U-rank $=1$, 
 this gives negative answer to 1 and 2.


Then note that in a supersimple theory there is no infinite forking chain.
As $\bigcup\_i tp(a\_i/A\_i)$ would not fork over some finite $B$ 
 and this would be contained in $A\_N$ for $N$ sufficiently big. 
Hence strictly superstable theories are a counter-example to question 3.


