Intersecting Families of Permutations

TL;DR

This paper proves that for large enough n, the largest k-intersecting families of permutations are cosets of stabilizers of k points, confirming a conjecture using eigenvalue and representation theory techniques.

Contribution

It establishes the structure of maximum k-intersecting permutation families for large n, confirming a conjecture of Deza and Frankl.

Findings

Largest k-intersecting families are cosets of stabilizers of k points.

Results hold for sufficiently large n depending on k.

Methods involve eigenvalue techniques and symmetric group representation theory.

Abstract

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.