Cardinal arithmetic and compactness phenomena in set theory

In his 1984 influential paper, G. Cantor discovered the existence of various types of infinites, setting the stage for the development of cardinal arithmetic — a core area of research in modern set theory. Another central theme in set theory is the study of compactness phenomena. Compactness is the phenomenon by which the properties of a mathematical structure (a group, a graph, a topological space, etc) are determined by the behavior of its small substructures. The primary focus of this colloquium is to examine the extent to which certain cardinal-arithmetic configurations influence the emergence of compactness phenomena. We shall begin the presentation providing a historical overview of the field of cardinal arithmetic culminating our exposition with the so-called Singular Cardinal Hypothesis (SCH). Later we will discuss compactness phenomena, putting the focus on two prominent combinatorial principles; namely, the Tree Property (TP) and Stationary Reflection (SR). Our discussion will be concluded with a presentation of a few recent results analyzing the interplay between the SCH, the TP and SR. Incidentally, those results answer two long-standing open problems by M. Magidor.