Mini-course (30.11.2023-25.01.2024, 6 lectures) - 3rd lecture:
We now deal with the more difficult task of forcing the converse inequalities on the left side of Cichon's diagram. We first show how Cohen reals add suitable Tukey connections (or what we also call "strong witnesses"). Next, we need to preserve such Tukey connections along FS iterations. For this purpose, we use a modern device of a preservation technique from Judah and Shelah (1990), and Brendle (1991), illustrating how combinatorial properties of forcing notions influence the preservation of such Tukey connections. However, serious problems related to the bounding number remain, which motivates more recent research that will be presented in the fourth lecture.