LAUSR

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 $n!/\text{poly}(n)$ , 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.