Adjacent-level matching in a Boolean lattice
= Adjacent-level matching in a Boolean lattice
For $1\leq i\leq n/2$, the inclusion graph between ranks $i-1$ and $i$ of $\mathcal P([n])$ has a matching saturating rank $i-1$. Indeed, each lower vertex has degree $n-i+1$, each upper vertex has degree $i$, and edge counting gives
$$
(n-i+1)|\mathcal A|\leq i|N(\mathcal A)|
$$
for every family $\mathcal A$ in the lower rank. This verifies the condition of the <Hall marriage theorem>.