A new proof for the Erdős–Ko–Rado theorem for the alternating group

Abstract

A subset S of the alternating group on n points is intersecting if for any pair of permutations π, σ in S, there is an element i ∈ {1, . . . , n} such that π(i) = σ(i). We prove that if S is intersecting, then |S| ≤ (n−1)! 2 . Also, we prove that if n ≥ 5, then the only sets S that meet this bound are the cosets of the stabilizer of a point of {1, . . . , n}.

An Erdős-Ko-Rado theorem for the group $$\hbox {PSU}(3, q)$$ PSU ( 3 , q )

“…Namely, we prove that the characteristic vector for any maximum intersecting set in PSU(3, q) is a linear combination of characteristic vectors of the canonical cliques in PSU (3, q). Showing the same result for other 2-transitive groups was a key result in proving that the groups have the strict-EKR property [1,10,15,16].…”

“…Although all 2-transitive groups have the ERK-property, it is not true that all 2-transitive groups have the strict-EKR property. It has been shown that Sym(n), Alt(n), PGL(2, q), PGL(3, q), PSL(2, q) [14] and the Mathieu groups all have the strict-EKR property by first showing that they have the EKR-module property [1,2,10,15,16,14]. It is shown in [17] that the Suzuki groups, Ree Groups, Higman-Sims, Symplectic groups also have the EKR-module property.…”

“…The result in each of [5,10,13] is that the symmetric group has the strict-EKR property. It has also been shown that many other groups also have the strict-EKR property [1,3,12,15,16]. It was recently shown that every 2-transitive group has the EKR property [17].…”

An Erdős–Ko–Rado theorem for finite 2-transitive groups

“…Recently there have been many papers proving that the natural extension of the Erdős-Ko-Rado theorem holds for specific permutation groups G (see [1,13,26,28,29,33]) and there are also two papers, [2] and [3], that consider when the natural extension of the Erdős-Ko-Rado theorem holds for transitive and 2transitive groups. Again, this means asking if the largest intersecting sets in G are the cosets in G of the stabiliser of a point.…”

The Erdős-Ko-Rado Property for Some 2-Transitive Groups

“…There have been several recent publications [2,6,12,15,16,17,19] that determine the maximum sets of elements from a permutation group such that any two permutations from the set both map at least one point to the same element. These results are considered to be versions of the Erdős-Ko-Rado (EKR) theorem [9] for permutations.…”