Almost tiling of the Boolean lattice with copies of a poset

Let P be a partially ordered set. If the Boolean lattice (2[n],⊂) can be partitioned into copies of P for some positive integer n, then P must satisfy the following two trivial conditions: (1) the size of P is a power of 2, (2) P has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if P only satisfies condition (2), we can still almost partition 2[n] into copies of P. We prove that if P has a unique maximal and minimal element, then there exists a constant c=c(P) such that all but at most c elements of 2[n] can be covered by disjoint copies of P.