  A large set of models beyond the Standard Model of particle physics suggest
that the top quark plays a special role in fundamental interactions. At the
same time some of these models predict that a particle responsible for dark
matter is present in the physical spectrum of the theory, and in particular
cases dark matter may be specially linked to the top sector. Such a new physics
configuration occurs, for instance, in composite frameworks that additionally
feature vector-like quarks as mediators of the interactions of the dark matter
with the Standard Model. We investigate the interplay of top-philic dark matter
and the presence of several vector-like top partners. Such a non-minimal setup
with several vector-like fermions is typical of composite scenarios, and we
make use of an effective model to study to which extent such non-minimality
modifies the collider and cosmological phenomenology of the minimal model, the
latter only including one vector-like mediator of the dark matter interactions
with the Standard Model.
