Combinatorial dichotomies in set theory

In this article we give an overview of a line of research in set theory that has reached a level of maturity and which, in our opinion, merits its being exposed to a more general audience. This line of research is concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered at the level of some other not so large cardinal, the most interesting being of course the level of the second uncountable cardinal. While this is of great interest to set theorists, one of the main forces behind this line of research stems from its applicability to other areas of mathematics where one encounters structures that are not necessarily countable, and therefore one is likely to encounter problems of this kind. We have split this overview into three parts each presenting a set theoretic combinatorial principle imposing a degree of compactness at some small cardinal. The three principles are natural dichotomies about chromatic numbers for graphs and about ideals of countable subsets of some index set. They are all relatively easy to state and apply and are therefore accessible to mathematicians working in areas outside of set theory. Each of these three dichotomies has its axiomatic as well as its mathematical side and we went into some effort to show the close relationship between these. For example, we have tried to show that an abstract analysis of one of these three set theoretic principles can sometimes lead us to results that do not require additional axioms at all but which could have been otherwise difficult to discover directly. We have also tried to select examples with as broad range as possible but their choices could still reflect a personal taste. We have therefore included an extensive reference list where the reader can find a more complete view on this area of current research in set theory. Finally we mention that while this article is meant for a larger audience which is typically interested in a general overview, we have tried to make the article also interesting to experts working in this area by including some of the technicalities especially if they appear to us as important tools in this area. For the same reason we will be pointing out a number of possible directions for further research. Our terminology and notation follows that of standard texts of set theory (see, for example, [53] and [67]). Recall that the Singular Cardinals Hypothesis, SCH, is the statement (∀θ) θ θ = θ · 2 . It is a strong structural assumption about the universe of sets as it answers all questions about the arithmetic of infinite cardinals. It is for this reason one of the most studied such hypotheses of set theory especially in the context of covering properties of inner models of set theory. Another set of principles which is even more relevant from this point of view are the square principles such as 2κ and 2(κ). Before we introduce these principles, recall that Cα (α < θ) is a C-sequence if Cα is a closed and unbounded subset of α for all α < θ. These combinatorial