A set of permutations A is said to contain a product if there are two permutations in it whose product also lies in A; otherwise A is called product free. We show that the density of a product free set A ⊆ An is at most O(1/√n). This result is asymptotically tight, and improves on earlier results of Gowers (O(1/n1/3)) and Eberhard (O(log7/2 n /√n)). We also show stability results. Our proof uses recent refinements of the hypercontractive inequality over Sn for the class of “global functions”, and in particular a version of the level 1 inequality in this setting. Based on a joint work with Peter Keevash and Noam Lifshitz.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
