Notes on the Dual Ramsey Theorem

The goal of these notes is to give a proof of the Dual Ramsey Theorem. This theorem was first proved in “A Dual Form of Ramsey’s Theorem” by Tim Carlson and Steve Simpson, Advances in Mathematics, 1984. The proof presented here is the same proof given in the paper and follows much of the same notation and terminology. The main change is that we will explicitly use the notion of “infinite variable words” as used by Simpson in later discussions of this theorem. (For example, Simpson uses this language in his discussion of reverse mathematics and the Dual Ramsey Theorem in the Proceedings Volume from the Boulder conference.) Before stating the Dual Ramsey Theorem, we need to introduce some notation and terminology. We let ω = {0, 1, . . .} and A be a finite (possibly empty) alphabet which is disjoint from ω.