We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF theory.

We survey the use of club guessing and other PCF constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal, element.

We describe a recipe for generating normal ideals on successors of singular cardinals. We show that these ideals are related to many weakenings of □ that have appeared in the literature. Our main purpose, however, is to provide an organized list of open questions related to these ideals.

This paper attempts to present and organize several problems in the theory of Singular Cardinals. The most famous problems in the area (bounds for the ℶ-function at singular cardinals) are well known to all mathematicians with even a rudimentary interest in set theory. However, it is less well known that the combinatorics of singular cardinals is a thriving area with results and problems that do not depend on a solution of the Singular Cardinals Hypothesis. We present here an annotated collection of representative problems with some references. Where the problems are novel, attribution is attempted and it is noted where money is attached to particular problems.

Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF theory. Important subthemes were mutual stationarity, Aronszajn trees, and superatomic Boolean Algebras.

The author notes considerable overlap between this paper and the unpublished report submitted to the Banff Center for the Workshop on Singular Cardinals Combinatorics, May 1–5, 2004.

We state some results related to the Silver Theorem.

We outline the proof of the consistency that there exists a saturated tail club guessing ideal on ω₁ which is not a restriction of the nonstationary ideal. A new class of forcing notions and the forcing axiom for the class are introduced for this purpose.

Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors.

In connection with some known results on uncountable cardinal sequences for superatomic Boolean algebras, we shall describe some open questions for superatomic Boolean algebras concerning singular cardinals.

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

We present a question about Suslin trees and the weak square hierarchy which was contributed to the list of open problems of the BIRS workshop.

We discuss some of the relationships between the notion of "mutual stationarity" of Foreman and Magidor and measurability in inner models. The general thrust of these is that very general mutual stationarity properties on small cardinals, such as the ℵns, is a large cardinal property. A number of open problems, theorems, and conjectures are stated.