Polyunsaturated posets and graphs and the Greene–Kleitman theorem Original Research Article

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 (J. Combin. Theory Ser. A 20 (1976) 41) proved that, for each k, every finite poset has a simultaneously k- and k+1-saturated chain partition. West (J. Combin. Theory Ser. A 41 (1986) 105) showed that the Greene–Kleitman Theorem is best possible in a strong sense by exhibiting, for each c⩾4, a poset with longest chain of cardinality c and no k- and l-saturated chain partition for any distinct, nonconsecutive k, l