Interval number of special posets and random posets Original Research Article

The interval number i(P) of a poset P is the smallest t such that P is a containment poset of sets that are unions of at most t real intervals. For the special poset Bn(k) consisting of the singletons and k-subsets of an n-element set, ordered by inclusion, i(Bn(k)) = min {k, n − k + 1} if |n/2 − k| ⩾ n/2 − (n/2)13. For bipartite posets with n elements or n minimal elements, i(P) ⩽ [n/(lgn − lglgn)] + 1. Finally, the fraction of the n-element posets having interval number between (1 − ε)n/8lg n and (32)([n/lg n − lglg n)] + 1) approaches 1 as n → ∞ (using the Kleitman-Rothschild model of random posets).