Sizes of flat maximal antichains of subsets

TL;DR

This paper explores the possible sizes of maximal antichains in the Boolean lattice, providing an alternative construction that demonstrates most sizes can be achieved with antichains of only two consecutive set sizes.

Contribution

It introduces an alternative construction method that shows almost all sizes of maximal antichains can be formed using only two consecutive levels of the lattice.

Findings

Most sizes of maximal antichains are obtainable with antichains of only two consecutive set sizes.

The paper characterizes the sizes of maximal antichains in the Boolean lattice.

Provides an alternative construction for maximal antichains.

Abstract

This is the second of two papers investigating for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only $l$-sets and $(l+1)$-sets for some $l$.