Fix some theory $T$.
Let $p$ be a type over some Model M and let $q$ be some global extension of $p$. 


Note:
The number of global coheirs of $p$ is bounded by the number of ultrafilters on $M$.


Also if $T$ is stable, it follows that $q$ is a non-forking extension of $p$ if and only if $q$ is a coheir of $p$.


If $T$ is simple and non-stable, then there is a type (say $p$) with a unbounded number of non-forking extension. Hence if $q$ is a non-forking global extension, then it does not need to be a coheir of $p$.


Is this true in general non-stable theories or for example NTP2-non-stable? Is there always a type $p$ over some model and some non-forking extension $q$ which is no coheir of $p$ 



Does non-stablity imply that there is a difference between non-forking and coheir extension
In general there are TP2 theories in which every global type non-forking over a small model is finitely satisfiable in it. An example of this is constructed in "On non-forking spectra", Artem Chernikov, Itay Kaplan, Saharon Shelah, <http://arxiv.org/abs/1205.3101>, Section 5.3.


Restricting to NTP2 theories, bounded number of non-forking extensions characterizes NIP, so in particular if every non-forking type is a coheir and T is NTP2, then it is NIP. See Section 4 in the paper above for a proof.


Finally, if T is a pseudofinite NIP theory, then a global type does not fork over a model M if and only if it is finitely satisfiable in it, see "A guide to NIP theories", Pierre Simon, Exercise 6.19. There are pseudofinite NIP theories that are not stable (e.g. the theory of a discrete linear order with the first and the last element - in this example it is easy to check this directly).


