Saxl graphs of primitive affine groups with sporadic point stabilizers

Abstract

Let [Formula: see text] be a permutation group on a set [Formula: see text]. A base for [Formula: see text] is a subset of [Formula: see text] whose pointwise stabilizer is trivial, and the base size of [Formula: see text] is the minimal cardinality of a base for [Formula: see text]. If [Formula: see text] has base size [Formula: see text], then the corresponding Saxl graph [Formula: see text] has vertex set [Formula: see text] and two vertices are adjacent if and only if they form a base for [Formula: see text]. A recent conjecture of Burness and Giudici states that if [Formula: see text] is a finite primitive permutation group with base size [Formula: see text], then [Formula: see text] has the property that every two vertices have a common neighbour. We investigate this conjecture in the case where [Formula: see text] is an affine group and a point stabilizer is an almost quasisimple group whose central quotient is either [Formula: see text] or [Formula: see text] for some sporadic simple group [Formula: see text]. We verify the conjecture for all but [Formula: see text] of the groups [Formula: see text].