 Solution

Let $\mathcal{A}\subseteq [n{]}^{(r)}$ be an intersecting family, with $n\ge 2r$. By the Iterated local LYM inequality, its upper shadow in level $n-r$ satisfiesThe family of complements ${\mathcal{A}}^c=\{[n]\setminus A:A\in \mathcal{A}\}$ also lies in level $n-r$ and has cardinality $|\mathcal{A}|$. It is disjoint from the upper shadow: if $A\subseteq [n]\setminus B$ for $A,B\in \mathcal{A}$, then $A\cap B=\varnothing$, contradicting intersection. Both families fit inside the $(n-r)$th level, soThus replacing the Kruskal-Katona theorem by Local LYM gives onlyThis agrees with the Erdős-Ko-Rado theorem when $n=2r$ but is weaker when $n>2r$.