Local LYM inequality

For $\mathcal{A}\subseteq [n{]}^{(r)}$, the local LYM inequality saysIt follows by counting pairs $(B,A)$ with $B\subset A$, $|B|=r-1$, and $A\in \mathcal{A}$.

The bipartite inclusion graph between consecutive nonempty ranks of the Boolean lattice is connected. Two rank-$r$ sets differing by one exchanged element share their rank-$(r+1)$ union as a neighbour; successive exchanges connect any two rank-$r$ sets. Every upper vertex has a lower neighbour. For the two middle ranks of an odd-dimensional Boolean lattice the graph is regular, so equality in the Local LYM inequality for a nonempty upper family forces the whole upper rank: equality admits no edges from its lower shadow to the remaining upper vertices.

Repeatedly applying the Local LYM inequality to upper shadows givesfor $\mathcal{A}\subseteq [n{]}^{(r)}$ and $0\le s\le n-r$. The analogous statement for iterated lower shadows follows directly from the stated lower-shadow form.