Maximal Flat Regular Antichains

Abstract

Let $2^{[m]}$ be ordered by set inclusion, and let $\mathcal{B}\subseteq 2^{[m]}$ be an antichain. An antichain $\mathcal{B}$ is called $k$-regular ($k\in \mathbb{N}$) if for each $i\in [m]$ there are exactly $k$ blocks $B_1,B_2,\dots ,B_k\in \mathcal{B}$ containing $i$. An antichain is called flat if there exists a positive integer $l$ such that $l\le |B|\le l+1$ for all $B\in \mathcal{B}$, and we call an antichain maximal if the collection of sets $\mathcal{B}\cup \{B\}$ is not an antichain for all $B\notin \mathcal{B}$. We call a maximal $k$-regular antichain $\mathcal{B}\subseteq (\genfrac{}{}{0ex}{}{[m]}{2})\cup (\genfrac{}{}{0ex}{}{[m]}{3})$ a $(k,m)$-MFRAC. In this paper we analyze $(k,m)$-MFRACs in the cases $m\le 7$, $k=m$, $k=m-1$, and $k=m-2$. We provide some constructions, give necessary conditions for existence, and mention some open problems.