Maximal antichains of subsets II: Constructions

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 $\left(\genfrac{}{}{0ex}{}{n}{\lceil n/2\rceil }\right)-\lceil n/2{\rceil }^2$ and the maximum size $\left(\genfrac{}{}{0ex}{}{n}{\lceil n/2\rceil }\right)$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.

This paper has been merged with arXiv:2106.02226