  A partition of a finite poset into chains places a natural upper bound on the
size of a union of k antichains. A chain partition is k-saturated if this bound
is achieved. Greene and Kleitman proved that, for each k, every finite poset
has a simultaneously k- and k+1-saturated chain partition. West showed that the
Greene-Kleitman Theorem is best-possible in a strong sense by exhibiting, for
each c \ge 4, a poset with longest chain of cardinality c and no k- and
l-saturated chain partition for any distinct, nonconsecutive k,l < c. We call
such posets polyunsaturated. We give necessary and sufficient conditions for
the existence of polyunsaturated posets with prescribed height, width, and
cardinality. We prove these results in the more general context of graphs
satisfying an analogue of the Greene-Kleitman Theorem. Lastly, we discuss
analogous results for antichain partitions.
