The Saxl graph of a permutation group

Let G be a permutation group on a set X and recall that a subset of X is called a base for G if its pointwise stabiliser is trivial. If G has a base of size 2, then we can associate a graph to G, with vertex set X and two points joined by an edge if they form a base. We call this the Saxl graph of G. In this talk, I will start with a brief introduction to bases, focussing on primitive groups and probabilistic methods. I will then introduce the Saxl graph and we will discuss some concrete examples and basic properties. I will finish by highlighting some recent results and open problems. This is joint work with Michael Giudici.