On the largest product-free subsets of the alternating groups

Abstract A subset A A of a group G G is called product-free if there is no solution to a=bc a = b c 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₍ S n is obtained by taking the set of all odd permutations, i. e. S₍ A₍ S n ∖ A n, where A₍ A n is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6 (2): 101–114, 1985) conjectured that the group A₍ 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₍ 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 \: \, (x) I, (I) I= \ 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₍ 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.