In the third talk, I will concentrate on inequalities for linear extensionsof finite posets.  I will start with several inequalities which do have a combinatorial proof.  I will then turn to Stanley's inequality and outline the proof why its defect does not have a combinatorialinterpretation.  Time permitting I will discuss what does that sayabout the equality cases and stability of the Alexandrov-Fenchelinequality. Joint work with Swee Hong Chan (STOC 2024).