Partitions of large Boolean lattices

AbstractLet 2n_ be the ordered set obtained from the Boolean lattice 2n by deleting both the greatest and the least elements. Define ƒ(n) to be the minimum number k such that there is a partition of 2n_ into k antichains of the same size except for at most one antichain of a smaller size. In the paper we examine the asymptotic behavior of ƒ(n) and we show that c1n⩽ƒ(n)⩽c2n2for some constants c1 and c2 and n sufficiently large. Moreover, we prove for all ordered sets P of size less than 5, a conjecture that for n sufficiently large there is a partition of 2n_ into ordered sets isomorphic to P if and only if some obvious divisibility conditions are satisfied