TY - JOUR

T1 - On the largest product-free subsets of the alternating groups

AU - Keevash, Peter

AU - Lifshitz, Noam

AU - Minzer, Dor

N1 - Publisher Copyright:
© The Author(s) 2024.

PY - 2024/9

Y1 - 2024/9

N2 - 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. Sn∖An, where An is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group An 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 An 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 An 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.

AB - 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. Sn∖An, where An is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group An 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 An 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 An 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.

UR - http://www.scopus.com/inward/record.url?scp=85194745774&partnerID=8YFLogxK

U2 - 10.1007/s00222-024-01273-1

DO - 10.1007/s00222-024-01273-1

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AN - SCOPUS:85194745774

SN - 0020-9910

VL - 237

SP - 1329

EP - 1375

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -