Ben Green's Open Problem 5

Which finite groups have the smallest biggest product-free sets?

We formalise this as: determine the supremum of exponents $\alpha$ such that every nontrivial finite group of order $n$ contains a product-free set of size $\geq c n^{\alpha}$ for some absolute constant $c > 0$. (The trivial group is excluded since its only product-free subset is empty.) Kedlaya [Ke97] showed that $\alpha = 11/14$ is admissible, and Green suggests this exponent may well be sharp; the candidate extremal family is the Ree groups ${}^2G_2(q)$, $q = 3^{2m+1}$.