Partitioning the Boolean lattice into chains of large minimum size

Let 2^[n] denote the Boolean lattice of order n, that is, the poset of subsets of (1,…, n) ordered by inclusion. Recall that 2^[n] may be partitioned into what we call the canonical symmetric chain decomposition (due to de Bruijn, Tengbergen, and Kruyswijk), or CSCD. Motivated by a question of Füredi, we show that there exists a function d(n) ∼ 1/2 n such that for any n ≥ 0, 2^[n] may be partitioned into (ⁿ_⌊n/2⌋) chains of size at least d(n). (For comparison, a positive answer to Füredi's question would imply that the same result holds for some d(n) ∼ π/2 n.) More precisely, we first show that for 0 ≤ j ≤ n, the union of the lowest j+1 elements from each of the chains in the CSCD of 2^[n] forms a poset T_j (n) with the normalized matching property and log-concave rank numbers. We then use our results on T_j(n) to show that the nodes in the CSCD chains of size less than 2d(n) may be repartitioned into chains of large minimum size, as desired.