Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric spaces), like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of topology, measure theory and functional analysis.
Lecture IV. Descriptive graph combinatorics.
This last lecture is about a relatively new subject, developed in the last two decades or so, which is at the interface of descriptive set theory and graph theory but also has interesting connections with other areas such as ergodic theory and probability theory. The object of study is the theory of definable graphs on Polish spaces and one investigates how combinatorial concepts, such as colorings and matchings, behave under definability constraints, i.e., when they are required to be definable or perhaps well-behaved in the topological or measure theoretic sense.