TITLE: Introduction to Infinitary Combinatorics
QUESTION [12 upvotes]: 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.

REPLY [4 votes]: 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.