Almost Tiling of the Boolean Lattice with Copies of a Poset

Abstract

Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\subset)$ 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 $P$.