Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size

Let 2^[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1,...,n} ordered by inclusion. Extending our previous work on a question of Füredi, we show that for any c>1, there exist functions e(n)∼ n/2 and f(n)∼c nlogn and an integer N (depending only on c) such that for all n>N, there is a chain decomposition of the Boolean lattice 2^[n] into (_⌊n/2⌋ ⁿ chains, all of which have size between e(n) and f(n). (A positive answer to Füredi's question would imply that the same result holds for some e(n)∼ π/2 n and f(n)=e(n)+1.) The main tool used is an apparently new observation about rank-collection in normalized matching (LYM) posets.