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

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 π, σ in S there is a point i ∈ {1, . . . , n} such that π(i) = σ(i). Deza and Frankl [9] proved that if S ⊆ S(n) is intersecting then |S| ≤ (n − 1)!. Further, Cameron and Ku [4] 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. * Research supported by NSERC.The Erdős-Ko-Rado theorem [7] is a central result in extremal combinatorics. There are many interesting proofs and extensions of this theorem, for a summary see [6]. The Erdős-Ko-Rado theorem gives a bound on the size of a family of intersecting k-subsets of a set and describes exactly which families meet this bound.1.1 Theorem. (Erdős, Ko and Rado [7]) Let k, n be positive integers with n > 2k. Let A be a family of k-subsets of {1, . . . , n} such that any two sets from A have non-trivial intersection, then |A| ≤ (n − 1)!. Moreover, |A| = (n−1)! if and only if A is the collection of all k-subsets that contain a fixed i ∈ {1, . . . , n}.The Erdős-Ko-Rado theorem has been extended to objects other than subsets of a set. For example, Hsieh [17] and Frankl and Wilson [8] give a version for intersecting subspaces of a vector space over a finite field, Berge [3] proves it for intersecting integer sequences, Rands [21] extends it to intersecting blocks in a design and Meagher and Moura [19] prove a version for partitions.The extension we give here is to intersecting permutations. Let S(n) be the symmetric group on {1, . . . , n}. Permutations π, σ ∈ S(n) are said to be intersecting if π(i) = σ(i) for some i ∈ {1, . . . , n}. Similar to the case for subsets of a set, there are obvious candidates for maximum intersecting systems of permutations, these are the sets S i,j = {π ∈ S(n) : π(i) = j}, i, j ∈ {1, . . . , n}.(1.1)These sets are the cosets of a stabiliser of a point.1.2 Theorem. (Cameron and Ku [4]) Let n ≥ 2. If S ⊆ S(n) is an intersecting family of permutations then:(a) |S| ≤ (n − 1)!.(b) if |S| = (n − 1)! then S is a coset of a stabiliser of a point.The proof given by Cameron and Ku uses an operation called fixing which is similar to the shifting operation used in the original proof of Erdős-Ko-Rado. They show that a maximum intersecting family of permutations is closed under this fixing operation. Assuming that the family contains the identity permutation, and thus each permutation in the family has a fixed point, they next consider the set system formed by the sets of fixed points for each permutation in the family. Cameron and Ku prove that if the family of permutations is closed

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