The Forbidden Cross Intersection Problem for Permutations

Nathan Keller; Noam Lifshitz; Ohad Sheinfeld

Paper

The Forbidden Cross Intersection Problem for Permutations

Abstract

We prove the following, for a universal constant

. Let

and

. Let

be families of permutations such that no

and

agree on exactly

values. Then

, with equality if and only if

, for some

. The range of values of

in the result is essentially optimal, as for any

, the statement fails for

and all

. This solves the cross-intersection variant of the Erdős-Sós forbidden intersection problem for permutations. The best previously known result, by Kupavskii and Zakharov (Adv.~Math., 2024), obtained the same assertion for

. We obtain our result by combining two recently introduced techniques: hypercontractivity of global functions and spreadness.