Abstract

In Part I, we provide a brief introduction to the notation and theory of the partition calculus. In particular, we explain several theorems and proofs which have some bearing on the results in Parts II, III, and IV. ;In Part II, we consider the Ramsey theory of non-special orders. We provide a very short and more elementary proof of the old result of P. Erdos and R. Rado that R→w+m,43 for each integer m and each real order type R. We then elaborate on some work of E. C. Milner and K. Prikry to prove a much stronger version of this result, namely that P→w+m,n3 for each pair of integers m and n and each non-special partial order P. We first prove this relation with P = o1 and then extend this proof to obtain the more general theorem. ;In Part III, we generalize to initial trees a few basic theorems of combinatorial set theory, including for example Ramsey's Theorem and a version of the Erdos-Rado Theorem. Along the way we develop the elementary theory of normal ideals on such trees. ;In Part IV, we extend a method of J. Baumgartner, A. Hajnal, and. S. Todorcevic to prove some new polarized partition relations: Two polarized versions of the Erdos-Rado Theorem and a related polarized partition relation involving weakly compact cardinals. Finally, we prove some results about finite polarized partition relations