A new proof of the Erdős–Ko–Rado theorem for intersecting families of permutations

Let S(n)S(n) be the symmetric group on nn points. A subset SS of S(n)S(n) is intersecting   if for any pair of permutations π,σπ,σ in SS there is a point i∈{1,…,n}i∈{1,…,n} such that π(i)=σ(i)π(i)=σ(i). Deza and Frankl [P. Frankl, M. Deza, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (3) (1977) 352–360] proved that if S⊆S(n)S⊆S(n) is intersecting then |S|≤(n−1)!|S|≤(n−1)!. Further, Cameron and Ku [P.J. Cameron, C.Y. Ku, Intersecting families of permutations, European J. Combin. 24 (7) (2003) 881–890] showed 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.