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∖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 {π:π(x)∈I,π(I)∩I=∅} 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.