The Saxl hypergraph of a permutation group

Given a permutation group G ≤ Sym(Ω), a subset B of Ω is said to be a base if its pointwise stabiliser in G is trivial, and thebase 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 extends this notion to any group for which b(G) ≥ 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.