Activities This Week

In this talk we describe two seemingly unrelated results on the symmetric group $S_n$. A family F of permutations is called t-intersecting if any two permutations in F agree on at least t values. In 1977, Deza and Frankl conjectured that for all $n>n_0(t)$, the maximal size of a t-intersecting subfamily of $S_n$ is $(n-t)!$. Ellis, Friedgut and Pilpel (JAMS, 2011) proved the conjecture for all $n>exp(exp(t))$ and conjectured that it holds for all $n>2t$. We prove that the conjecture holds for all $n>ct$ for some constant c. A well-known problem asks for characterizing subsets A of $S_n$ whose square $A^2$ contains (=”covers”) the alternating group $A_n$. We show that if A is a union of conjugacy classes of density at least $exp(-n^{2/5-\epsilon})$ then $A_n \subset A^2$. This improves a seminal result of Larsen and Shalev (Inventiones Math., 2008) who obtained the same with 1/4 in the double exponent. The common feature of the two results is the main tool we use in the proofs, which is (perhaps surprisingly) analytic - hypercontractive inequalities for global functions. We shall discuss the new tool (introduced recently by Keevash, Lifshitz, Long and Minzer, JAMS 2024) and other directions in which it may be applied.

Based on joint works with Noam Lifshitz, Dor Minzer, and Ohad Sheinfeld