SOME RESULTS ABOUT HINDMAN'S THEOREM AND GOWERS' THEOREM IN REVERSE MATHEMATICS.

Abstract

This thesis focuses on two theorems related to combinatorics: Hindman's Theorem and Gowers' Theorem (which can be regarded as a generalization of Hindman's Theorem). They are both about the coloring on a semigroup: giving a coloring function on a certain semigroup, there exists an infinite subset whose closure under semigroup operations is monochromatic.

For Hindman's Theorem, we use a variant of Blass, Hirst and Simpson's method to construct some coloring functions so that no Delta_3 set can be the a solution to these colorings for Hindman's Theorem; this is about the lower bound of Hindman's Theorem. For Gowers' Theorem, we use the principle of Mathias forcing to prove Gowers' Theorem in a subsystem of second order arithmetic; this is an initial step about the upper bound of Gowers' Theorem.