Polyunsaturated posets and graphs and the Greene-Kleitman theorem (Q1850016)

A partition of a poset \(P\) into chains is called \(k\)-saturated if its size is maximum. As was proved by P. West, for each \(c\geq 4\) there exists a poset of height \(c\) in which no partition into chains is simultaneously \(k\)- and \(\ell\)-saturated, here \(k\), \(\ell\) are any two distinct non-consecutive integers both less than \(c\); such a poset is called polyunsaturated. The paper shows a construction of such graphs with the smallest possible width and cardinality. Then graph theory is applied. The concept of a strong Greene-Kleitman graph (SGK graph) is introduced and used for the study of posets.