The length of random subsets of Boolean lattices

Ž . w x 4 ABSTRACT: We form the random poset P n, p by including each subset of n s 1, . . . , n with probability p and ordering the subsets by inclusion. We investigate the length of the Ž . longest chain contained in P n, p . For pGern we obtain the limit distribution of this random variable. For smaller p we give thresholds for the existence of chains which imply Ž . Ž . that almost surely the length of P n, p is asymptotically n log ny log log 1rp rlog 1rp. Q 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 177]194, 2000