Introduction to Infinitary Combinatorics

What are some good texts for someone interested in becoming acquainted with the "big ideas" of infinitary combinatorics? If you'd like more specificity, assume the reader has respectable mathematical maturity and a working knowledge of finitary combinatorics and set theory.

Solution 1:

The following is strictly combinatorics, leaving aside many issues related to independence results. First of all, I do not know of any reference that covers all topics I can think of off the top of my head, so instead, I will present a list.

You are actually lucky, since a nice historical overview of infinitary combinatorics was just published. It is the chapter "Infinite combinatorics" by Jean Larson, in the "Handbook of the History of Logic", volume 6. It does not quite present everything, since its emphasis is on the 20th century so some of the latest developments are missing.

A good quick overview of themes up to 30-odd years ago is in K. Kunen's chapter ("Combinatorics") in the "Handbook of mathematical logic". Chapter II of his book on Set Theory is also a decent introduction, and several topics not developed in the chapter are presented in the exercises.

I would then suggest to at least skim through Williams "Combinatorial set theory" and Erdős-Hajnal-Mate-Rado "Combinatorial set theory". The emphasis on the latter is definitely partition calculus (Ramsey theory of infinite sets) but quite a few other topics are presented as well.

After this, it is a bit harder to find a more up-to-date comprehensive treatment. There is Hajnal "Infinite combinatorics", chapter 42 in the "Handbook of combinatorics". And the "Handbook of set theory", of course, covers a few of these topics in detail: There are chapters on stationary sets (Jech), partition relations (Hajnal and Larson), coherent sequences (Todorcevic, who also wrote a book on the subject, and a couple of books on set theoretic Ramsey theory), cardinal characteristics (a chapter by Blass and one by Bartoszynski; this is a topic that was missing from the previous books and chapters, except for Larson's chapter mentioned on the second paragraph), cardinal arithmetic (Abraham-Magidor), and singular cardinal combinatorics (Eisworth, there are also a couple of excellent articles on this topic by Cummings, that you can find on his webpage).

There is a nice recent book by Halbeisen, "Combinatorial set theory", that covers some of these topics, and discusses Martin's axiom and some combinatorics in the absence of choice.

Another nice reference is the "Handbook of set theoretic topology". Of particular interest towards infinitary combinatorics are the chapter on trees (by Todorcevic) and on the Proper forcing axiom (by Baumgartner). On the topic of forcing axioms, there is much more to say, but you won't find it in book form (this is partly my fault, as I am co-authoring a book on this, which should have appeared ages ago).

Except for their inclusion in Larson's chapter mentioned in the second paragraph above, there are several big topics missing so far: The key reference for large cardinals is "The higher infinite", by Kanamori. This book also has an introduction to determinacy, but this topic requires knowledge of descriptive set theory to be appreciated properly. A quick introduction (a bit outdated now) to some of the combinatorics under determinacy is the book "Infinitary combinatorics and the axiom of determinateness" by Kleinberg. Other bits can be seen through the relevant chapters in the Handbook of set theory.

Solution 2:

There are several different things that could be called "infinitary combinatorics". Andres Caicedo has listed several good texts for what I would call set-theoretic combinatorics. I will list two or three that concern infinitary Ramsey theory, such as Szemeredi's theorem and Hindman's theorem.

The classic Recurrence in Ergodic Theory and Combinatorial Number Theory by Furstenberg is a very nice exposition of recurrence methods (topological and ergodic) for infinitary combinatorics. The material here would be a prerequisite for understanding the Green-Tao theorem on arithmetical progressions in the primes. I have not had the opportunity to look at the more recent book Additive Combinatorics by Tao and Vu.

The book Algebra in the Stone-Čech Compactification by Hindman and Strauss describes a completely different approach to infinitary combinatorics, via ultrafilters on $\mathbb{N}$ and other discrete semigroups.