Almost tiling of the Boolean lattice with copies of a poset

TL;DR

This paper proves that for posets with a unique maximal and minimal element, the Boolean lattice can be nearly partitioned into copies of the poset, covering all but a bounded number of elements.

Contribution

It extends previous results by showing near-partitions are possible for a broader class of posets satisfying only the extremal element condition.

Findings

Almost partition of Boolean lattice into copies of P

Bounded number of uncovered elements

Generalization of previous conjecture

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 $P$.