Partition calculus is the area of set theory that deals with Ramsey theory; it is devoted to Ramsey’s theorem and its infinite and infinitary generalizations. This means both strengthenings of Ramsey’s theorem for sets of natural numbers (like the Carlson-Simpson or the Galvin-Prikry theorems characterizing the completely Ramsey sets in terms of the Baire property) and for larger cardinalities (like the -Rado theorem), as well as variations in which the homogeneous sets are required to possess additional structure (like the Baumgartner-Hajnal theorem).
Ramsey theory is a vast area and by necessity we won’t be able to cover (even summarily) all of it. There are many excellent references, depending on your particular interests. Here are but a few:
- Paul András Hajnal, Attila Máté, Richard Rado, Combinatorial set theory: partition relations for cardinals, North-Holland, (1984).
- Ronald Graham, Bruce Rothschild, Joel Spencer, Ramsey theory, John Wiley & Sons, second edn., (1990).
- Neil Hindman, Dona Strauss, Algebra in the Stone- compactification, De Gruyter, (1998).
- Stevo High-dimensional Ramsey theory and Banach space geometry, in Ramsey methods in Analysis, Spiros Argyros, Stevo Birkhäuser (2005), 121–257.
- András Hajnal, Jean Larson, Partition relations, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., to appear.
I taught a course on Ramsey theory at Caltech a couple of years ago, and expect to post notes from it at some point. Here we will concentrate on infinitary combinatorics, but I will briefly mention a few finitary results.