The Saxl hypergraph of a permutation group
Melissa Lee, Anthony Pisani
TL;DR
This work extends the Saxl graph paradigm to Saxl hypergraphs $\mathcal{H}(G)$, whose edges are the bases of size $b(G)$ for a permutation group $G$. It provides a complete classification of when $\mathcal{H}(G)$ is complete, linking this to Frobenius groups and a range of almost simple and classical group actions, including $\mathrm{PSL}_2$/$\mathrm{PGL}_2$ on projective lines and Suzuki-type actions. The authors generalise the Common Neighbour Conjecture to hypergraphs, establish edge-disjoint variants under broad hypotheses, and develop the concept of gossip numbers to measure neighbourhood richness, illustrating both positive results and explicit counterexamples. They also introduce and partially classify flag-spanning tours, showing how valency and group structure constrain the existence of such tours, particularly for base sizes $b(G)\in\{3,4\}$. Overall, the paper advances understanding of base-structure-driven hypergraphs in permutation group theory and connects these combinatorial objects to classical group actions and transitivity properties with potential implications for computational and combinatorial applications.
Abstract
Given a permutation group $G \le \mathrm{Sym}(Ω)$, a subset $B$ of $Ω$ is said to be a base if its pointwise stabiliser in $G$ is trivial, and the base size $b(G)$ is the minimum size of a base. In the notable case $b(G) = 2$, Burness and Giudici define the Saxl graph of $G$ to be the graph on $Ω$ with bases of size 2 as edges. Later work of Freedman et al. extends this notion to any group for which $b(G) \ge 2$, taking the pairs of points contained in bases of size $b(G)$ for edges. We study an alternative generalisation, the Saxl hypergraph, where bases of size $b(G)$ are themselves the edges. In particular, we consider groups with complete Saxl hypergraphs, primitive groups whose Saxl hypergraphs have flag-spanning tours, and appropriate generalisations of Burness and Giudici's Common Neighbour Conjecture.