I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017.
Title: Strong colorings and their applications.
Abstract. Consider the following questions.
- Is the product of two $\kappa$-cc partial orders again $\kappa$-cc?
- Does there exist a regular hereditary separable topological space which is non-Lindelof?
- Given an $\aleph_1$-sized Abelian group $(G,+)$, must there exist a unary function $f:G\rightarrow G$ such that any proper substructure of $(G,+,f)$ be countable?
It turns out that all of the above questions can be decided (in one way), provided that there exists a certain “strong coloring” (or “wild partition”) of a corresponding uncountable graph.
In this tutorial, we shall present some of the techniques involved in constructing such strong colorings, and demonstrate how partial orders/topological spaces/algebraic structures may be derived from these colorings.