Ron Graham 
Rudiments of Ramsey Theory [PDF ebook] 

In every sufficiently large structure which has been partitioned there will always be some well-behaved structure in one of the parts. This takes many forms. For example, colorings of the integers by finitely many colors must have long monochromatic arithmetic progressions (van der Waerden’s theorem); and colorings of the edges of large graphs must have monochromatic subgraphs of a specified type (Ramsey’s theorem). This book explores many of the basic results and variations of this theory. Since the first edition of this book there have been many advances in this field. In the second edition the authors update the exposition to reflect the current state of the art. They also include many pointers to modern results.