Some graph theoretic results associated with Ramsey's theorem

We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y)≤cy^x-1log log y/logy for x≥3.