In this series of two talks, we present recent progress in the structure theory of combinatorial objects at the level of the cardinal $\aleph_2$ and above. We specifically focus on the behavior of directed sets, linear orders, trees, semi-lattices, gaps, and towers.

The first part of the series introduces the Transitive List Dichotomy, which is defined for every successor of a regular cardinal. This dichotomy yields several significant consequences regarding the "specialization" of these objects.

In the second part, we analyze the structure of directed sets with cofinality below $\aleph_\omega$ under the Tukey order.

The first part represents joint work with Stevo Todorčević and Boriša Kuzeljević.