Intersecting families of permutations

Authors:
Peter J. Cameron;C. Y. Ku
Affiliations:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK;School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
Venue:
European Journal of Combinatorics
Year:
2003

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Abstract

Let Sn be the symmetric group on the set X = {1, 2,...,n}. A subset S of Sn is intersecting if for any two permutations g and h in S, g(x) = h(x) for some x ∈ X (that is g and h agree on x). Deza and Frankl (J. Combin. Theory Ser. A 22 (1977) 352) proved that if S ⊆ Sn is intersecting then |S| ≤ (n - 1)!. This bound is met by taking S to be a coset of a stabiliser of a point. We show that these are the only largest intersecting sets of permutations.