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Let $G$ be a permutation group on a set $\Omega$. A base for $G$ is a subsetof $\Omega$ whose pointwise stabiliser is trivial, and the base size of $G$ isthe minimal cardinality of a base. If $G$ has base size $2$, then thecorresponding Saxl graph $\Sigma(G)$ has vertex set $\Omega$ and two verticesare adjacent if they form a base for $G$. A recent conjecture of Burness andGiudici states that if $G$ is a finite primitive permutation group with basesize $2$, then $\Sigma(G)$ has the property that every two vertices have acommon neighbour. We investigate this conjecture when $G$ is an affine groupand a point stabiliser is an almost quasisimple group whose unique quasisimplesubnormal subgroup is a covering group of a sporadic simple group. We verifythe conjecture under this assumption, in all but ten cases.

Saxl graphs of primitive affine groups with sporadic point stabilisers