On the largest product-free subsets of the alternating groups

• 成果类型: Article
• 署名作者: Keevash, Peter; Lifshitz, Noam; Minzer, Dor
• 署名单位: University of Oxford; Hebrew University of Jerusalem; Massachusetts Institute of Technology (MIT)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01273-1
• 发表日期: 2024
• 页码: 1329-1375
• 关键词: intersecting families

摘要

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 Sn 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 Sos (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 {pi : pi(x) is an element of I, pi(I) boolean AND I = empty set} 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.