## Advanced Topics in Mathematical Logic (V5A6)

### Two cardinal combinatorics

### Lecturer

- Dr. Bernhard Irrgang

### Time and location

Tuesdays, 10 - 12, SR CThursdays, 10 - 12, SR C

### Contents

The lecture course provides an introduction to results and problems in two cardinal combinatorics. It is a common phenomenon that infinite (and finite) mathematical structures are characterized by two cardinals. Typical examples are: Kurepa trees (height and number of braches), Suslin trees (height and size of antichains), partitions (size of partitioned set and size of homogeneous subset), superatomic Boolean algebras (height and width), topological spaces (various cardinal characteristics). But there are many more! The question is for which values of the two cardinal such structures exist. In some cases it is known what possibilities are consistent, however, in many cases we do not even know with which method we should attack the problem. This will be discussed in the course. Special emphasis will be put on my method of higher-dimensional forcing. Planned topics are:1. Introduction

2. Gap-1 Morasses

3. Higher-dimensional forcing which preserves GCH

4. Higher-dimensional forcing which destroys GCH

5. Gap-2 morasses

6. Spread and size of Hausdorff spaces

7. On long chains in P(omega) mod finite.