The Erdős-Ko-Rado theorem for the derangement graph of the projective general linear group acting on the projective space

Abstract

In this paper we prove an Erd\H{o}s-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group $\mathrm{PGL}_{n+1} (q)$, in its natural action on the points of the $n$-dimensional projective space, is either a coset of the stabiliser of a point or a coset of the stabiliser of a hyperplane. This gives a positive solution to (K.~Meagher, P.~Spiga, An Erd\H{o}s-Ko-Rado theorem for the derangement graph of $\mathrm{PGL}(2, q)$ acting on the projective line, \textit{J. Comb. Theory Series A} \textbf{118} (2011), 532-544.).

On the intersection density of the symmetric group acting on uniform subsets of small size

“…Using Sagemath, we obtained the intersection density of some of these 4-homogeneous groups in the next table. For the group G = PΓL 2 (32) acting on the 4-subsets of [33], the intersection density can be computed by considering the maximum cliques in the complement of the derangement graph Γ G . The largest and least eigenvalues of Γ G are 1023 and −33, respectively.…”

“…However, there are 2-transitive groups such as PGL 3 (q) acting on the projective plane [27] that do not have the strict-EKR property. See [1,2,3,8,23,26,33] for other examples concerning EKR and/or strict-EKR properties.…”

Simple Algebraic Proofs of Uniqueness for Erdős-Ko-Rado Theorems

“…The latter often requires laborious calculations involving ranks of certain submatrices of M ′ and ad-hoc arguments concerning the 0/1 structure of such submatrices. This technique is quite versatile in that it has been shown to work not only in combinatorial settings but also algebraic ones such as P GL(n, q) [34]. The trade off here is that the method seems to require lots of case analyses and ad-hoc arguments that result in longer more complicated proofs.…”

On complete multipartite derangement graphs

“…The EKR theorem has been well studied and generalized for numerous combinatorial objects in the past 50 years [6,7,8,11,13,19,20,22,25]. Of interest to us is the generalization of Theorem 1.1 for the symmetric group by Deza and Frankl in [7].…”

“…A big step toward this classification is the result of Meagher, Spiga and Tiep [20], which says that every finite 2-transitive group has the EKR property. More examples of primitive groups having the EKR property are given in [1,2,5,9,17,19,22].…”