(Joint seminar with set theory group)

The study of compactness phenomena at uncountable cardinals has been a central line of research in combinatorial set theory since the mid-twentieth century. In the first part of this talk, we will give a broad overview of this area of research and survey some of its most prominent results. We will then look at a few recent results concerning compactness phenomena for uncountable graphs. We first look at possible generalizations of the de Bruijn-Erdos compactness theorem for chromatic numbers to uncountable cardinalities, in particular showing that consistently there are large uncountable graphs witnessing extreme failures of compactness for, e.g., the property of having a countable chromatic number. We then turn to the study of the structure of the collections of finite subgraphs of uncountably chromatic graphs, answering a question of Erdos, Hajnal, and Szemeredi about the growth rates of chromatic numbers in such collections of... more

The study of compactness phenomena at uncountable cardinals has been a central line of research in combinatorial set theory since the mid-twentieth century. In the first part of this talk, we will give a broad overview of this area of research and survey some of its most prominent results. We will then look at a few recent results concerning compactness phenomena for uncountable graphs. We first look at possible generalizations of the de Bruijn-Erdos compactness theorem for chromatic numbers to uncountable cardinalities, in particular showing that consistently there are large uncountable graphs witnessing extreme failures of compactness for, e.g., the property of having a countable chromatic number. We then turn to the study of the structure of the collections of finite subgraphs of uncountably chromatic graphs, answering a question of Erdos, Hajnal, and Szemeredi about the growth rates of chromatic numbers in such collections of... more