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We form the random poset 𝒫(n, p) by including each subset of [n]={1,…,n} with probability pand ordering the subsets by inclusion. We investigate the length of the longest chain contained in 𝒫(n, p). For p≥e/nwe obtain the limit distribution of this random variable. For smaller pwe give thresholds for the existence of chains which imply that almost surely the length of 𝒫(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

We form the random poset P(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 P(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 P(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