Let $G$ be a permutation group on a set $\Omega$. A base for $G$, is a subset $B$ of $\Omega$ such that the pointwise stabiliser of the elements of $B$ is trivial. There has been a large amount of recent research on the size of a base of a primitive permutation group, culminating with the recent proof of Pyber’s Conjecture. At the same time there has been a large amount of work devoted to finding the primitive groups with a base of size two. For such groups we can define the Saxl graph of $G$ to be the graph with vertex set $\Omega$ and two elements are joined by an edge if they are a base. I will discuss some recent work with Tim Burness that investigates some of the properties of this graph.

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