Maximal antichains of subsets II: Constructions

TL;DR

This paper proves that all smaller integers than a certain threshold are achievable as sizes of maximal antichains in the Boolean lattice, completing the characterization of possible maximal antichain sizes.

Contribution

It establishes that every integer smaller than a specific bound can be realized as a size of a maximal antichain in the Boolean lattice.

Findings

All smaller m are sizes of maximal antichains.

Characterization of possible sizes of maximal antichains.

Completes the classification for the second part of the sequence.

Abstract

This is the second in a sequence of three papers investigating the question 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 previous paper we characterized those $m$ between $\binom{n}{\lceil n/2\rceil}-\lceil n/2\rceil^2$ and the maximum size $\binom{n}{\lceil n/2 \rceil}$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.