An approximate isoperimetric inequality for r-sets

Abstract: We prove a vertex-isoperimetric inequality for [n]^r, the set of all r-element subsets of {1,2,...,n}, where x,y \in [n]^r are adjacent if |x \Delta y|=2. Namely, if \mathcal{A} \subset [n]^r 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).