Proofs of Ramsey's Theorem and a combinatorial approach to determining bounds for Ramsey numbers

Ramsey's Theorem is about order in large sets. The infinite version of Ramsey's Theorem states that an infinite set X has an infinite subset Y such that all the members of Y satisfy a given combinatorial relation. The finite version states that for any natural number n there exists a least integer M(n) such that if a set S has M(n) members it will have a subset T with n members and all of the members of T will satisfy a given combinatorial relation. Several proofs of the theorem are presented, and some methods for determining bounds of Ramsey Numbers are explored, along with some results obtained using the various methods. The emphasis is on combinatorial methods. There is also a short discussion of the field of study which came to be known as Ramsey Theory and few notes on some Ramsey type problems.