An approximate isoperimetric inequality for r-sets

We prove a vertex-isoperimetric inequality for [n]^®, the set of all r-element subsets of {1,2,…,n}, where x,y \in [n]^® are adjacent if |x \Delta y|=2. Namely, if $\mathcal{A}$ \subset [n]^® with |$\mathcal{A}$|=\alpha {n \choose r}, then the vertex-boundary b($\mathcal{A}$) satisfies |b($\mathcal{A}$)| \geq c\sqrt{\frac{n}{r(n-r)}} \alpha(1-\alpha) {n \choose r}, where c is a positive absolute constant. For \alpha bounded away from 0 and 1, this is sharp up to a constant factor (independent of n and r).