On the Topic of Ramsey Theory

Ramsey theory studies conditions that are necessary so that we may preserve 'order.' Often, we will start with a large structure, and we will break it into finitely many classes. Our typical theorems will say that if the initial structure is large enough, then one of the classes must contain a copy of the original structure. Usually, we will find the smallest number of elements required so that this property is guaranteed to occur; this is often a difficult, but illuminating, problem.