A Simpli ed Guide to Large Antichains in thePartition

Let n denote the lattice of partitions of an n-set, ordered by reenement. We show that for all large n there exist antichains in n whose size exceeds n 1=35 S (n; K n). Here S (n; K n) is the largest Stirling number of the second kind for xed n, which equals the largest rank within n. Some of the more complicated aspects of our previous proof of this result are avoided, and the variance of a certain random variable Z which plays a key role in the construction is determined to within O(1). 1 The problem. Let S be a nite set with n elements. A partition of S is a collection of pairwise disjoint subsets of S, called blocks, whose union is S. We say that partition 1 reenes partition 2 , denoted 1 < 2 , provided 1 is obtained from 2 by further partitioning one or more blocks of 2. Under the reenement relation the set n of all partitions of an n-set is a partially ordered set (poset), in fact, a lattice, and the problem is to nd large antichains in this poset. An antichain in a poset is a collection of elements no two of which are related. How shall we decide if an antichain in the partition lattice n is \large" in the absence of a proven upper bound? Observe that when 1 < 2 the partition 1 has more blocks than 2. Thus the set of all partitions having a xed number of blocks, say k blocks, is an antichain. The size of this antichain is S(n; k), the Stirling number of the second kind; we let S(n; K n) denote the largest Stirling number of the second kind. It has been known since 1] that there exist antichains within n which for all large n are strictly larger than S(n; K n). However, all such antichains constructed to date have cardinality asymptotically equal to S(n; K n). It is the purpose of this paper to construct antichains A n such that jAj=S(n; K n) ! 1. The important ideas for this construction can be traced to the 1985 work 4]. The latter gave a heuristic, though not rigorously established, argument that antichains 1