Ramsey-type principles, regressivity and well-orderings

Different variations of Ramsey's Theorem and Hindman's Theorem are studied by relating them to Well-Ordering Preserving Principles through both direct implications over RCA₀ and via computable and Weihrauch reductions. A regressive version of Hindman's Theorem is then introduced as a restriction of Taylor's Canonical Hindman's Theorem, and several results concerning the Reverse Mathematics of this Regressive Hindman's Theorem and its natural restrictions are provided. These results include the equivalence over RCA₀ of Arithmetical Comprehension with the first non-trivial restriction of the principle. Finally, an ordinal analysis of the class of theories RCA₀ + WO(δ) is carried out for any ordinal δ satisfying ω·δ=δ, thereby providing a measure of the strength of all such systems, whose relevance stems from their connections to a number of well-studied combinatorial principles.