Some Results in Combinatorial Set Theory

In this thesis, we present a number of results in combinatorial set theory, especially in Ramsey theory and its variations, compactness principles and dimension
theory. Chapter 2 concerns the tail cone version of the Halpern-Lauchli theorem at uncountable cardinals. Relative to large cardinals, we establish the consistency of a
tail cone version of the Halpern-Lauchli theorem at a large cardinal, which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other
applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation. We elaborate on how this method is helpful in separating various partition relations on generalized rational numbers.