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

Let $G$ be a transitive permutation group on a set $\Omega$, and suppose $G_{\alpha}\cap G_{\beta}=1$ for some distinct $\alpha, \beta\in\Omega$. The Saxl graph $\Sigma(G)$ of $(G, \Omega)$ is defined as the graph with vertex set $\Omega$, where two vertices $\alpha', \beta'$ are adjacent if and only if $G_{\alpha'}\cap G_{\beta'}=1$. Burness and Giudici conjectured that for every primitive permutation group $G$, its Saxl graph has the property that any two vertices share a common neighbor. We focus on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, $soc(G)\in \{PSL(2,q),PSU(3,q), Ree(q),Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been treated in two earlier papers. The purpose of the present paper is to settle the case $soc(G)=PSU(3,q)$. To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas.