These notes, which are based on a graduate course and seminars given at Wes-
leyan University in the fall of 1997, contain many of the main results of chromatic
and density Ramsey theory in Z. This subfield of combinatorial number theory
deserves attention due in part to the fact that its principal theorems are (i)
natural, (ii) easy to formulate, and (iii) non-trivial. These three features are
hallmarks of the best in pure mathematics.
Most of the results we treat were originally proved combinatorially. We
concentrate here on a variety of alternative approaches proceeding by way of
topological dynamics and ergodic theory. Our goal is not to prove theorems of
the greatest generality in the shortest amount of time, but to allow the reader
to observe the methodology gradually unfolding.
The problems we will be attacking fall roughly into three classes, which are
perhaps best introduced by example. As it happens, one result in each category
stands out as characteristic.