Common-neighbour property for almost simple and diagonal primitive groups

Establish that every primitive permutation group of almost simple or diagonal type with base size two has the property that any two vertices of its Saxl graph have a common neighbour.

Background

The paper disproves the previously proposed common-neighbour conjectures for primitive groups of arbitrary base size and, in particular, for base-two groups in the affine, product, and twisted wreath O’Nan–Scott types. Consequently, only the almost simple and diagonal types remain as possible types in which the original base-two conjecture might hold.

Computational searches found no counterexamples of non-affine type up to degree 10^8 and none of diagonal type up to degree 10^24. Together with earlier positive results for several families, these observations motivate the amended conjecture that the common-neighbour property continues to hold for all primitive groups of almost simple or diagonal type with base size two.