On the largest product-free subsets of the alternating groups

A subset $A$ $A$ of a group $G$ $G$ is called product-free if there is no solution to $a=bc$ $a=bc$ with $a,b,c$ $a,b,c$ all in $A$ $A$ . It is easy to see that the largest product-free subset of the symmetric group $S_n$ $S_{n}$ is obtained by taking the set of all odd permutations, i.e. $S_n\mathrm{\setminus }{A}_n$ $S_{n} \backslash A_{n}$ , where $A_n$ $A_{n}$ is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group $A_n$ $A_{n}$ also contains a product-free set of constant density. This conjecture was refuted by Gowers (whose result was subsequently improved by Eberhard), still leaving the long-standing problem of determining the largest product-free subset of $A_n$ $A_{n}$ wide open. We solve this problem for large $n$ $n$ , showing that the maximum size is achieved by the previously conjectured extremal examples, namely families of the form $\{\pi :\pi (x)\in I,\pi (I)\cap I=\mathrm{\varnothing }\}$ $\left \{ \pi :\,\pi (x)\in I, \pi (I)\cap I=\varnothing \right \} $ and their inverses. Moreover, we show that the maximum size is only achieved by these extremal examples, and we have stability: any product-free subset of $A_n$ $A_{n}$ of nearly maximum size is structurally close to an extremal example. Our proof uses a combination of tools from Combinatorics and Non-abelian Fourier Analysis, including a crucial new ingredient exploiting some recent theory developed by Filmus, Kindler, Lifshitz and Minzer for global hypercontractivity on the symmetric group.