Common neighbour conjectures for Saxl graphs fail at every base size

Abstract

For a finite permutation group, a base is a set of points with trivial pointwise stabiliser, and the generalised Saxl graph records which pairs of points lie together in a base of minimum size. Burness and Giudici conjectured that any two vertices of the Saxl graph of a primitive group of base size two have a common neighbour, and Freedman, Huang, Lee and Rekvényi extended this conjecture to arbitrary base size. We disprove both. For each integer $B\ge2$ we construct infinitely many primitive groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size two, where this is the usual Saxl graph, we obtain three further infinite families, one each of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan--Scott types; in the affine and product type families the Saxl graphs have diameter exactly three. This answers Problem~21.29 in the Kourovka Notebook in the negative. In the positive direction, we prove the Burness--Giudici conjecture for every primitive affine group whose point stabiliser is almost quasisimple of sporadic type, completing work of Lee and Popiel. We conjecture that no base-two counterexample of almost simple or diagonal type exists.