Restricted Ramsey configurations

Abstract

In recent years there has been great development in the area becoming known as Ramsey Theory. In this paper we explore some restricted Ramsey theorems, best illustrated by the following beautiful result of NeSetIil and RodI. We begin with the first nontrivial example of the original theorem of Ramsey: Given any two-coloring of the edges of KB there exists a monochromatic triangle. Let us use the notation H--f (G), if any c-coloring of the edges of a graph H yields a monochromatic G. In this instance we write K, -+ (K& . It was asked, by P. Erdos, what graphs H have the property H--f (K& , and whether such H exist if you restrict the clique number w(H). In an elegant, though complex, paper Folkman [4] showed that there exist H satisfying H---f (K& , where w(H) = 3. A full generalization, using a totally different method, was given by NeSetIil and Rod1 [6], who showed that for all G, c there exists a graph H so that H -+ (G), and w(H) = w(G).