Polyunsaturated Posets and Graphs and the Greene-Kleitman Theorem

Abstract

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.