On the Saxl graphs of primitive groups with soluble stabilisers (Q2097198)

If \(G\) is a transitive permutation group on a finite set \(\Omega\), a base for \(G\) is a subset of \(\Omega\) with trivial pointwise stabiliser. The base size of \(G\), denoted \(b(G)\), is the minimal size of a base. In the case of \(b(G)=2\) the authors define the Saxl graph \(\Sigma(G)\) of \(G\), which has vertex set \(\Omega\) and two vertices are adjacent if and only if they form a base. A number of open problems concerning Saxl graphs are presented in [\textit{T. C. Burness} and \textit{M. Giudici}, Math. Proc. Camb. Philos. Soc. 168, No. 2, 219--248 (2020; Zbl 1479.20006)] and the aim of this paper is to address some of these questions for the permutation groups in the collection \(\mathcal{G}\) of finite almost simple primitive groups \(G\) with \(b(G)=2\) and soluble point stabilisers. Furthermore, let \(\mathcal{L}\) be the subcollection of \(\mathcal{G}\) of groups \(G \in \mathcal{G}\) with \(G_{0} = L_{2}(q)\) and \(H\) of type \(\mathrm{GL}_{1}(q) \wr S_{2}\) or \(\mathrm{GL}_{1}(q^{2})\) where \(G_{0}\) denotes the socle of \(G\) and \(H\) is a point stabiliser. The reviewer reports some of the main results obtained in this paper. Theorem 1.1: Let \(G \leq \mathrm{Sym}(\Omega)\) be a permutation group in \(\mathcal{G}\). Then any two vertices in \(\Sigma(G)\) have a common neighbour. In particular, \(\Sigma(G)\) has diameter 2. The clique number of \(\Sigma(G)\) is denoted by \(\omega(G)\) and the independence number by \(\alpha(G)\). Theorem 1.3: Let \(G \leq \mathrm{Sym}(\Omega)\) be a permutation group in \(\mathcal{G}\) with socle \(G_{0}\) and assume that either \(G \not \in \mathcal{L}\) or \(G \leq PGL_{2}(q)\). Then \(\omega(G) \geq 4\), with equality if and only if \(G = A_{5}\) and \(\Omega\) is the set of 2-element subsets of \(\{1,\ldots, 5\}\). Theorem 1.5: Let \(G \leq \mathrm{Sym}(\Omega)\), \(G \in \mathcal{G}\). Then either \(\alpha(G) \geq 4\), or \(G = A_{5}\), \(\Omega\) is the set of 2-element subsets of \(\{1, \ldots, 5 \}\) and \(\alpha(G) = 2\).