$(m-1)$-Regular Antichains on [m]

Abstract

Let $\mathcal{B}\subseteq 2^{[m]}$ be an antichain of size $|\mathcal{B}|=:n$. $2^{[m]}$ is ordered by inclusion. An antichain $\mathcal{B}$ is called $k$-regular ($k\in \mathbb{N}$), if for each $i\in [m]$ there are exactly $k$ sets $B_1,B_2,\dots ,B_k\in \mathcal{B}$ containing $i$. In this case, we say that $\mathcal{B}$ is a $(k,m,n)$-antichain.

Let $m\ge 2$ be an arbitrary natural number. In this note, we show that an $(m-1,m,n)$-antichain exists if and only if $n\in [m+2,(\genfrac{}{}{0ex}{}{m}{2})–2]\cup \{m,(\genfrac{}{}{0ex}{}{m}{2})\}$.