An Erdös--Ko--Rado Theorem for the Derangement Graph of ${PGL}_3(q)$ Acting on the Projective Plane

Abstract

In this paper we prove an Erdös--Ko--Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group ${PGL}_3(q)$, in its natural action on the points of the projective line, is either a coset of the stabilizer of a point or a coset of the stabilizer of a line. This gives the first evidence for the veracity of Conjecture 2 from K. Meagher and P. Spiga, An Erdös-Ko-Rado Theorem for the Derangement Graph of ${PGL}(2,q)$ Acting on the Projective Line [J. Combin. Theory Ser. A, 118 (2011), pp. 532--544].