A subset of a group is said to be product free if it does not contain the product of two elements in it. We consider how large can a product free subset of $A_n$ be?
In the talk we will completely solve the problem by determining the largest product free subset of $A_n$.
Our proof combines a representation theoretic argument due to Gowers, with an analytic tool called hypercontractivity for global functions. We also make use of a dichotomy between structure and a pseudorandomness notion of functions over the symmetric group known as globalness.
Based on a joint work with Peter Keevash and Dor Minzer.