New constructions and bounds for nonabelian Sidon sets with applications to Turán-type problems

Abstract An S k $S_k$ upper S Subscript k - set in a group Γ $\Gamma $ normal upper Gamma is a set A ⊆ Γ $A\subseteq \Gamma $ upper A subset of or equal to normal upper Gamma such that α 1 … α k = β 1 … β k $\alpha _1\dots \alpha _k=\beta _1\dots \beta _k$ alpha 1 ellipsis alpha Subscript k Baseline equals beta 1 ellipsis beta Subscript k Baseline with α i , β i ∈ A $\alpha _i,\beta _i\in A$ alpha Subscript i Baseline comma beta Subscript i Baseline element of upper A implies ( α 1 , … , α k ) = ( β 1 , … , β k ) $(\alpha _1,\ldots ,\alpha _k)=(\beta _1,\ldots ,\beta _k)$ left parenthesis alpha 1 comma ellipsis comma alpha Subscript k Baseline right parenthesis equals left parenthesis beta 1 comma ellipsis comma beta Subscript k Baseline right parenthesis . An