From graphs to groups: Tuple regularity and $k$-ultrahomogeneity

Tuple regularity and $k$-ultrahomogeneity are refinements of the classical notion of ultrahomogeneity, which is a well-studied symmetry notion of graphs, or, more generally, of relational structures. For many graph classes, all of these concepts are well-understood.

Ultrahomogeneity has also been studied for groups. In particular, the ultrahomogeneous finite groups were classified by Li (1999) and, independently, by Cherlin and Felgner (2000). However, especially in order to study finite groups from a combinatorial perspective, we are interested in refinements of this notion for groups.

In this talk, I will discuss the concepts of $k$-ultrahomogeneity and $k$-tuple regularity for various graph classes, and define suitable analogs for groups. At the end, I will briefly discuss a classification result for finite groups.

Parts of this talk are based on joint work with Irene Heinrich.