Homotopical combinatorics is an emerging field that studies combinatorial structures encoding aspects of equivariant homotopy theory, equivariant algebra, and abstract homotopy theory. Central to this field is the concept of transfer systems, originally defined to encode the homotopy theory of $N_\infty$-operads, which govern multiplicative structures in equivariant stable homotopy theory. Transfer systems also play a crucial role in controlling the structure of bi-incomplete Tambara functors, fundamental objects in equivariant algebra, and in encoding model structures on posets.

In this talk, we define transfer systems and explore their compatibility. I will present recent results on compatible pairs of transfer systems for the group $(G = C_{p^r q^s})$, identifying conditions under which these systems form only trivially compatible pairs. If time permits, we will also explore how transfer systems can be used to identify model structures on finite posets.