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Consider the $S_n$-action on $\mathbb{R}^n$ given by permuting coordinates.This paper addresses the following problem: compute $\max_{v,H} |H\cap S_nv|$as $H\subset\mathbb{R}^n$ ranges over all hyperplanes through the origin and$v\in\mathbb{R}^n$ ranges over all vectors with distinct coordinates that arenot contained in the hyperplane $\sum x_i=0$. We conjecture that for $n\geq3$,the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove thatif $p$ is the largest prime with $p\leq n$, then $\max_{v,H} |H\cap S_nv|\leq\frac{n!}{p}$. In particular, this proves the conjecture when $n$ or $n-1$ isprime.

What fraction of an $S_n$-orbit can lie on a hyperplane?