An $S_k$-set in a group $\Gamma$ is a set $A\subseteq\Gamma$ such that $\alpha_1\cdots\alpha_k=\beta_1\cdots\beta_k$ with $\alpha_i,\beta_i\in A$ implies $(\alpha_1,\ldots,\alpha_k)=(\beta_1,\ldots,\beta_k)$. An $S_k'$-set is a set such that $\alpha_1\beta_1^{-1}\cdots\alpha_k\beta_k^{-1}=1$ implies $\alpha_i=\beta_i$ or $\beta_i=\alpha_{i+1}$ for some $i$. We give constructions of large $S_k$-sets in the groups $\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)$ and of large $S_2$-sets in $\mathrm{Sym}(n)\times\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)\times\mathrm{Alt}(n)$. A probabilistic bound for `nice' groups obtains large $S_2'$-sets in $\mathrm{Sym}(n)$. We also give various upper bounds; in particular, if $k$ is even and $\Gamma$ has a normal abelian subgroup with bounded index then any $S_k$-set has size at most $(1-\varepsilon)|\Gamma|^{1/k}$.
   
 We describe some connections between $S_k$-sets and extremal graph theory. We determine up to a constant factor the minimum outdegree in a digraph with no subgraph in $\{C_{2,2},\ldots,C_{k,k}\}$, where $C_{\ell,\ell}$ is the orientation of $C_{2\ell}$ with two maximal directed $\ell$-paths. Contrasting with undirected cycles, the extremal minimum outdegree for $\{C_{2,2},\ldots,C_{k,k}\}$ is much smaller than that for any $C_{\ell,\ell}$. We count the directed Hamilton cycles in one of our constructions to improve the upper bound for a problem on Hamilton paths introduced by Cohen, Fachini, and Körner.

This talk is based on joint work with Michael Tait; see https://arxiv.org/abs/2509.07750.