Interval number of special posets and random posets

The interval number i(P) of a poset P is the smallest t such that P is a containment of sets that are unions of at most t real intervals. For the special poset Bn(k) consisting of the singletons and ksubsets of an n-element set, ordered by inclusion, i(Bn(k)) = min {k, n − k + 1} if |n/2 − k | ≥ n/2 − (n/2). 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 8lgn and (3/2)( n lgn − lglgn  + 1) approaches 1 as n → ∞ (i.e., this involves the Kleitman-Rothschild model of random posets). Ke ywords: poset, representation parameter, interval number, containment representation, dimension Running head: INTERVAL NUMBER OF POSETS 1Research supported in part by NSA/MSP Grant MDA904-90-H-4011.