The Saxl hypergraph of a permutation group

TL;DR

This paper introduces the Saxl hypergraph, a new combinatorial structure associated with permutation groups, and explores its properties, including cases with complete hypergraphs and connections to existing conjectures.

Contribution

It generalizes the Saxl graph to hypergraphs, studies their properties for various groups, and relates to conjectures on common neighbors in these hypergraphs.

Findings

Identified conditions for complete Saxl hypergraphs.

Analyzed primitive groups with flag-spanning tours.

Extended the Common Neighbour Conjecture to hypergraphs.

Abstract

Given a permutation group $G \le \mathrm{Sym}(\Omega)$, a subset $B$ of $\Omega$ 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 $\Omega$ 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.