Obtaining Bounds for Ramsey Numbers

There are two equivalent problem statements for the Ramsey number n = R(k; l). n is the minimum number of vertices in the graph such that it contains a complete graph of k vertices, or an independent set of l vertices. n is the minimum number of vertices such that if all the edges of the complete graph on n vertices, denoted by K n is colored with two colors, fRed, Blueg, then there exists a Red K k or a blue K l. The above problem statement can be generalized in terms of the number of colors used to color the graph, as well as the deenition of an edge. When we generalize the number of colors used to color the graph (the previous deenition used a two-coloring), we deene the Ramsey number as The above Ramsey number gives the minimum number of vertices n such that if the edges of K n are colored with j colors, there exists a monochromatic K r i , 1 i j. When we generalize the number of vertices that deenes an edge, we deene the Ramsey number as This gives the minimum number of vertices in a s-uniform complete hypergraph, such that if the hyperedges are colored with j colors, there exists a monochromatic complete hypergraph of r i , 1 i j vertices. In this report, we study Ramsey numbers, in particular R(k; l). In the next section, we sketch a brief history of Ramsey theory. Section 3 focuses on techniques used to obtain upper and lower bounds for Ramsey numbers. The upper bound for Ramsey numbers is obtained easily using the pigeonhole principle. To obtain the lower bound, we study two techniques-the probabilistic method and the reenement method. 2 History of Ramsey Theory Ramsey theory was introduced originally by Ramsey in 1930. At present there are several branches of study within Ramsey Theory, for example, we have Graph Ramsey Theory, Euclidean Ramsey Theory and Ramsey numbers. This report focuses on Ramsey numbers. There has been signiicant 1