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 √ n log n 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 ⌊n/2⌋ ) 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.