  We introduce a new two-sided stable matching problem that describes the
summer internship matching practice of an Australian university. The model is a
case between two models of Kamada and Kojima on matchings with distributional
constraints. We study three solution concepts, the strong and weak stability
concepts proposed by Kamada and Kojima, and a new one in between the two,
called cutoff stability. Kamada and Kojima showed that a strongly stable
matching may not exist in their most restricted model with disjoint regional
quotas. Our first result is that checking its existence is NP-hard. We then
show that a cutoff stable matching exists not just for the summer internship
problem but also for the general matching model with arbitrary heredity
constraints. We present an algorithm to compute a cutoff stable matching and
show that it runs in polynomial time in our special case of summer internship
model. However, we also show that finding a maximum size cutoff stable matching
is NP-hard, but we provide a Mixed Integer Linear Program formulation for this
optimisation problem.
