David Ellis

Families of permutations with a forbidden intersection

A family of permutations is said to be 't-intersecting' if any two permutations in the family agree on at least t points. It is said to be (t-1)-intersection-free if no two permutations in the family agree on exactly t-1 points. Deza and Frankl conjectured in 1977 that a t-intersecting family of permutations in S_n can be no larger than a coset of the stabiliser of t points, provided n is large enough depending on t; this was proved by the speaker and independently by Friedgut and Pilpel in 2008. We give a new proof of a stronger statement: namely, that a (t-1)-intersection-free family of permutations in S_n can be no larger than a coset of the stabiliser of t points, provided n is large enough depending on t. This can be seen as an analogue for permutations of seminal results of Frankl and Furedi on families of k-element sets. Our proof is partly algebraic and partly combinatorial; it is more 'robust' than the original proofs of the Deza-Frankl conjecture, using a combinatorial 'quasirandomness' argument to avoid many of the algebraic difficulties of the original proofs. Its robustness allows easier generalisation to various other permutation groups. Based on joint work with Noam Lifshitz (Hebrew University of Jerusalem).