The length of random subsets of Boolean lattices

We form the random poset (n, p) by including each subset of [n]={1,…,n} with probability p and ordering the subsets by inclusion. We investigate the length of the longest chain contained in (n, p). For p≥e/n we obtain the limit distribution of this random variable. For smaller p we give thresholds for the existence of chains which imply that almost surely the length of (n, p) is asymptotically n(log n−log log 1/p)/log 1/p. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 177–194, 2000