New constructions and bounds for nonabelian Sidon sets with applications to Tur\'an-type problems

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 that there exists $i$ such that $\alpha_i=\beta_i\text{ or }\beta_i=\alpha_{i+1}$. We give explicit constructions of large $S_k$-sets in the group $S_n$ and $S_2$-sets in $S_n\times S_n$ and $A_n\times A_n$. We give probabilistic constructions for `nice' groups which obtain large $S_2$-sets in $A_n$ and $S_2'$-sets in $S_n$. We also give upper bounds on the size of $S_k$-sets in certain groups, improving the trivial bound by a constant multiplicative factor. We describe some connections between $S_k$-sets and extremal graph theory. In particular, we determine up to a constant factor the minimum outdegree of a digraph which guarantees even cycles with certain orientations. As applications, we improve the upper bound on Hamilton paths which pairwise create a two-part cycle of given length, and we show that a directed version of the Erd\H{o}s-Simonovits compactness conjecture is false.