Abstract

Let G be a permutation group on a finite set Ω. A base for G is a subset of Ω with trivial pointwise stabiliser, and the base size of G is the minimal size of a base for G. This classical invariant has been studied intensively since the early years of group theory in the 19th century, finding a wide range of applications.

In 2020, Burness and Giudici defined the Saxl graph of G with base size 2, whose vertices are the points of Ω, and where two vertices are adjacent if and only if they form a base for G. Later in the same year, I reported on my study on Saxl graphs at the Discrete Mathematics Seminar of SUSTech, which was my undergraduate thesis and eventually turned to a joint paper with Jiyong Chen.

Here at the same place, after my PhD thesis submission, I will review the main open problems and new development of this direction over the past four years, highlighting a recent generalisation with Saul Freedman, Melissa Lee and Kamilla Rekvényi.