On the largest product-free subsets of the alternating groups

A subset \(A\) of a group \(G\) is called product-free if there is no solution to \(a=bc\) with \(a,b,c\) all in \(A\). It is easy to see that the largest product-free subset of the symmetric group \(S_{n}\) is obtained by taking the set of all odd permutations, i.e. \(S_{n} \backslash A_{n}\), where \(A_{n}\) is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group \(A_{n}\) 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 \(A_{n}\) wide open. We solve this problem for large \(n\), showing that the maximum size is achieved by the previously conjectured extremal examples, namely families of the form \(\left \{ \pi :\,\pi (x)\in I, \pi (I)\cap I=\varnothing \right \} \) 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 \(A_{n}\) 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.