Ramsey theory

In this report we will take a look at various proofs of Ramsey's theorem, some of the bounds that result from those proofs and applications of Ramsey's theorem. We will consider the proof of Ramsey himself, the proof of Skolem, the proof given by Erd\H{o}s and Szekeres and the proof of again Erd\H{o}s and Rado. The best upper bound for higher...