4 problems

Burness–Giudici conjecture. Every two vertices of $\Sigma(G)$ have a common neighbour.

Let $G\leq \operatorname{Sym}(\Omega)$ be a finite primitive permutation group with base size $b(G)\geq 2$, and let $\mathcal{H}(G)$ be its Saxl hypergraph. Exceptional-groups conj…

Let $G\leq \operatorname{Sym}(\Omega)$ be a primitive permutation group with base size $b(G)=2$, and let $\Sigma(G)$ be its Saxl graph, whose vertices are the points of $\Omega$ an…

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite primitive permutation group with base size $b(G)\geqslant 2$. For $\alpha\in\Omega$, call a $G_\alpha$-orbit on $\Omega$ almo…