  Modern spin-foam models of four dimensional gravity are based on a discrete
version of the $Spin(4)$ Plebanski formulation. Beyond what is already in the
literature, we clarify the meaning of different Plebanski sectors in this
classical discrete model. We show that the linearized simplicity constraints
used in the EPRL and FK models are not sufficient to impose a restriction to a
single Plebanski sector, but rather, three Plebanski sectors are mixed. We
propose this as the reason for certain extra `undesired' terms in the
asymptotics of the EPRL vertex analyzed by Barrett et al. This explanation for
the extra terms is new and different from that sometimes offered in the
spin-foam literature thus far.
