发表机构
King’s College London; Heilbronn Institute for Mathematical Research; Warwick Mathematics Institute, University of Warwick（伦敦国王学院; 希尔伯恩数学研究所; 华威大学华威数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Saxl图的公共邻点猜想在所有基大小下均不成立，构造了对应反例并否定回答了《Kourovka笔记》的相关问题，同时完成了该猜想在特定仿射群下的成立性证明。
AI 中文摘要
对于有限置换群，基是使得逐点稳定子群为平凡的点集，广义Saxl图记录了哪些点对共同属于最小大小的基。Burness与Giudici猜想，基大小为2的本原群的Saxl图中任意两个顶点都有一个公共邻点，Freedman、Huang、Lee和Rekvényi将该猜想推广到任意基大小的情形。我们证明这两个猜想均不成立：对于每个整数B≥2，我们构造了无限多个基大小为B的本原群，其广义Saxl图包含两个不相邻且无公共邻点的顶点。在基大小为2的情形（即通常的Saxl图），我们得到了另外三个无限族，分别属于仿射型、乘积型和扭曲圈积型，因此该猜想在O'Nan–Scott五种类型中的三种下不成立；在仿射型和乘积型族中，Saxl图的直径恰好为3。这否定地回答了《Kourovka笔记》中的问题21.29。在肯定方向上，我们证明了对于每个点稳定子群为散在型几乎拟单群的本原仿射群，Burness–Giudici猜想成立，完成了Lee与Popiel的工作。我们猜想不存在几乎单型或对角型的基大小为2的反例。
英文摘要
For a finite permutation group, a base is a set of points with trivial pointwise stabiliser, and the generalised Saxl graph records which pairs of points lie together in a base of minimum size. Burness and Giudici conjectured that any two vertices of the Saxl graph of a primitive group of base size two have a common neighbour, and Freedman, Huang, Lee and Rekvényi extended this conjecture to arbitrary base size. We disprove both. For each integer $B\ge2$ we construct infinitely many primitive groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size two, where this is the usual Saxl graph, we obtain three further infinite families, one each of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan--Scott types; in the affine and product type families the Saxl graphs have diameter exactly three. This answers Problem~21.29 in the Kourovka Notebook in the negative. In the positive direction, we prove the Burness--Giudici conjecture for every primitive affine group whose point stabiliser is almost quasisimple of sporadic type, completing work of Lee and Popiel. We conjecture that no base-two counterexample of almost simple or diagonal type exists.
Comments14 pages
