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 $|T|$?


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.


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