“Poly-unsaturated" posets: the Greene-Kleitman theorem is best possible
Journal of Combinatorial Theory Series A
Some sequences associated with combinatoral structures
Discrete Mathematics
Discrete Mathematics - Special volume (part two) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs” (“The theory of regular graphs”)
Graph Theory With Applications
Graph Theory With Applications
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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 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.