We talked about extremal problems for set systems: collections of subsets of an element sets, - Sperner’s theorem, the Erdos-Ko-Rado theorem, and quite a few more. (See here, here and here.) What happens when we consider collections of permutations rather than collection of sets? Not so much is known on extremal problems for families of permutations.
David Ellis, Ehud Friedgut, and Haran Pilpel have recently proved an old conjecture of Deza and Frankl regarding an analog for permutations of the Erdos-Ko-Rado Theorem. They showed that for every if is sufficiently large, any family of permutations on an -element set such that every two permutations in the family agree in at least places, contains at most permutations.
The proof uses spectral methods and representations of the symmetric group.