The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,q)

Let $G$ be a transitive permutation group on $\mathrm{\Omega }$ with two points $\alpha ,\beta \in \mathrm{\Omega }$ such that $G_{\alpha }\cap G_{\beta }=1$. The Saxl graph $\mathrm{\Sigma }(G)$ of the pair $(G,\mathrm{\Omega })$ is the graph with vertex set $\mathrm{\Omega }$, while two vertices ${\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }}$ are adjacent if and only if $G_{{\alpha }^{\mathrm{\prime }}}\cap G_{{\beta }^{\mathrm{\prime }}}=1$. It was conjectured by Burness and Giudici that the Saxl graph $\mathrm{\Sigma }(G)$ of any primitive permutation group $G$ has the property that any two vertices have a common neighbor. We focused on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$, that is, those with $soc(G)\in \{PSL(2,q),PSU(3,q),Ree(q),Sz(q)\}$. The case of $soc(G)=PSL(2,q)$ has been published in two papers. This paper will address most cases where $soc(G)=PSU(3,q)$, with the exception of a particularly intricate configuration in which the point stabilizer contains $PSO(3,q)$. That specific configuration has been treated in a separate paper.