Improved bounds on the partitioning of the Boolean lattice into chains of equal size

The Boolean lattice 2[n]2[n] is the power set of [n][n] ordered by inclusion. If cc is a positive integer, a cc-partition of a poset is a chain partition, where all but at most one of the chains have size cc. We prove that if n=Ω(c2)n=Ω(c2), then 2[n]2[n] has a cc-partition. This improves a theorem of Lonc.We also prove a generalization of this result. If cc is a positive integer and PP is a poset whose comparability graph is connected, then PnPn has a cc-partition if nn is sufficiently large.