Original title: Common neighbour conjectures for Saxl graphs fail at every base size
Aluna Rizzoli, Adam R. Thomas · September 2026
Mathematicians study groups, which are collections of symmetries that move points around. In these groups, a baseA set of points in a permutation group that uniquely identifies each group element when fixed. is a small set of points that locks every movement in place. A Saxl graphA graph where vertices are points of a group, and edges connect pairs that can be part of a minimum-size base. connects any two points that can share a smallest possible base. Past researchers conjectured that any two points in these graphs would always share a common neighbourIn a graph, a vertex that is connected to both vertices of a given pair.. The authors proved this guess wrong. They showed that these shared neighbours do not always exist. For every smallest base size from two upwards, they built infinite families of counterexamples containing points with no shared neighbour. For base size two, they found failures across three different group structures, answering an open math problem. They also proved the original rule does hold for one specific group family, and predicted two other types never fail.
Imagine a sports team where each player has specific roles, and a small core group is needed to cover all positions. If you pick two players who aren't both in the same core, you might assume there's another player who could team up with either of them to form a valid core. In this paper, the team represents a mathematical group, the players are points, and the core is a base. The authors show that in some team structures, there are pairs of players with no such linking player, breaking the assumption.
This disproves a conjecture that mathematicians thought held for all primitive permutation groups, changing the understanding of how these groups are structured. It affects researchers in group theory by showing that a key property fails in multiple cases, guiding future work.
The paper does not provide counterexamples for almost simple or diagonal types in base size two; it only conjectures that none exist. It also focuses on primitive groups, so the results may not apply to other group types. The proof for certain affine groups shows that the conjecture holds in some cases, so it is not universally false.
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Common neighbour conjectures for Saxl graphs fail at every base size — Aluna Rizzoli, Adam R. Thomas
Abstract licence: CC0
Explanation generated by Synthesis from the abstract — it is not the original paper. This is a preprint, so it hasn't necessarily been peer reviewed.