Hechler forcing and its relatives
This thesis is divided into two main parts. In the first part, we focus on analyzing the properties of the single step extension Hechler forcing, as well as two closely related forcing notions which we refer to as the tree Hechler forcing and the non-decreasing Hechler forcing.
We prove that Hechler forcing and tree Hechler forcing are inequivalent forcing notions. This entails proving a representation theorem for the dominating reals in the Hechler extension. As corollaries, we answer a question due to Laflamme and settle in the negative a conjecture of Brendle and Loewe. We prove a strengthening of an unpublished result of Goldstern, showing that the product of any two forcing notions which add a dominating real will add a Hechler real. We prove that every subextension of the Hechler extension contains a Cohen real. We prove (jointly with Itay Neeman) that Hechler forcing and the non-decreasing Hechler forcing are equivalent as forcing notions.
In the second part of the thesis, we study the interaction of polychromatic Ramsey theory and monochromatic Ramsey theory in a variety of settings. The main theme of our work in this part is that the monochromatic theory is strictly stronger than the polychromatic theory. We prove that some amount of the axiom of choice is necessary to prove the rainbow Ramsey theorem. We also prove that the rainbow Ramsey theorem is not sufficiently strong as a choice principle to imply Ramsey's theorem. We prove (jointly with Anush Tserunyan) that rainbow Ramsey flavored infinite exponent partition relations conflict with the axiom of choice.
We investigate the (countable) combinatorial power of the rainbow Ramsey theorem; to accomplish this we introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the classical Ramsey ultrafilters. We investigate the relationship between rainbow Ramsey ultrafilters and other well-known types of special ultrafilters which encapsulate various combinatorial principles. For example, we prove that every rainbow Ramsey ultrafilter is nowhere dense, and that there exists a rainbow Ramsey ultrafilter which is not discrete. Finally, we give several cardinal characteristics of the continuum new characterizations in the spirit of polychromatic Ramsey theory.