An approximate isoperimetric inequality for r-sets

We prove a vertex-isoperimetric inequality for [n]((r)), the set of all r-element subsets of {1, 2, ... , n}, where x, y is an element of [n]((r)) are adjacent if vertical bar x Delta y vertical bar - 2. Namely, if A subset of [n]((r)) with vertical bar A vertical bar = alpha(n r), then its vertex-boundary b(A) satisfies vertical bar b(A)vertical bar >= c root n/r(n - r) alpha(1 - alpha) ((n) (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).