On some Ramsey like statements

This thesis investigates combinatorial principles from order theory and Ramsey theory with a focus on foundational aspects within the framework of reverse mathematics and computability theory. The first part focuses on order dimension theory, a classical topic at the intersection of order theory and combinatorics. Intuitively, the dimension measures how “far” a poset is from being linearly ordered. We are interested in statements that give an upper bound to the dimension of a poset in terms of the dimension of its subposets, obtained by removing one or more points. We analyze these bounding theorems, calibrating their logical strength within subsystems of second order arithmetic. The second part examines the notion of strong indivisibility, a particular Ramsey-like property. We focus on countable structures, with particular emphasis on Cameron’s classification theorem of strongly indivisible graphs. We study this classification from the perspectives of reverse mathematics and computable combinatorics, investigating the role of induction axioms and effective constructions. In the final part we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. Historically, largeness notions were associated to countable ordinals and systems of fundamental sequences. To extend this approach we develop a more flexible framework using blocks and barriers. Since the complexity of barriers can be measured by countable ordinals, we define and study Ramsey ordinals, a generalization of the well known Ramsey numbers of classical finite Ramsey theory.