  We prove that Boolean functions on $S_n$, whose Fourier transform is highly
concentrated on irreducible representations indexed by partitions of $n$ whose
largest part has size at least $n-t$, are close to being unions of cosets of
stabilizers of $t$-tuples. We also obtain an edge-isoperimetric inequality for
the transposition graph on $S_n$ which is asymptotically sharp for subsets of
$S_n$ of size $n!/\textrm{poly}(n)$, using eigenvalue techniques. We then
combine these two results to obtain a sharp edge-isoperimetric inequality for
subsets of $S_n$ of size $(n-t)!$, where $n$ is large compared to $t$,
confirming a conjecture of Ben Efraim in these cases.

Combinatorics