A new proof of the Erd\H{o}s-Ko-Rado theorem for intersecting families of permutations

TL;DR

This paper presents a novel proof of the Erd ext{"o}s-Ko-Rado theorem for intersecting families of permutations, establishing that maximum intersecting sets are cosets of stabilizers, with a different approach from previous proofs.

Contribution

The paper introduces a new proof technique for the Erd ext{"o}s-Ko-Rado theorem in the context of permutation groups, differing from existing methods.

Findings

Maximum intersecting families are cosets of stabilizers of a point.

The new proof offers an alternative approach to the classical result.

The bound on the size of intersecting families is reaffirmed.

Abstract

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations \pi, \sigma in S there is a point i in {1,...,n} such that \pi(i)=\sigma(i). Deza and Frankl \cite{MR0439648} proved that if S a subset of S(n) is intersecting then |S| \leq (n-1)!. Further, Cameron and Ku \cite{MR2009400} show that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result.