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An isoperimetric inequality for conjugation-invariant sets in the symmetric group

We prove an isoperimetric inequality for conjugation-invariant sets of size $k$ in $S_n$, showing that these necessarily have edge-boundary considerably larger than some other sets of size $k$ (provided $k$ is small). Specifically, let $T_n$ denote the Cayley graph on $S_n$ generated by the set of all transpositions. We show that if $A \subset S_n$ is a conjugation-invariant set with $|A| = pn! \leq n!/2$, then the edge-boundary of $A$ in $T_n$ has size at least $$c \cdot \frac {\log_2 (\tfrac 1{p})}{\log_2 \log_2 (\tfrac 2{p})}\cdot n \cdot |A|,$$ where $c$ is an absolute constant. (This is sharp up to an absolute constant factor, when $p = Θ(1/s!)$ for any $s \in \{1,2,...,n\}$.) It follows that if $p = n^{-Θ(1)}$, then the edge-boundary of a conjugation-invariant set of measure $p$ is necessarily a factor of $Ω(\log n / \log \log n)$ larger than the minimum edge-boundary over all sets of measure $p$.