A graph with many vertices cannot be homogeneous, i.e., for any pair of integers (i,j) all large graphs must contain either a complete subgraph on i vertices or an independent set of size j. The Ramsey number for (i,j) is the smallest integer R such that all graphs with at least R vertices have this property. For example, the (3,3) Ramsey number is 6; if a graph has 6 or more vertices, then is must contain a triangle or an independent set of size 3. The (4,4) Ramsey number is 18, found in 1954 [GG] . The (5,5) Ramsey number is still unknown; it is between 43 and 52. This thesis deals with subgraphs slightly different from the classical types. The subgraphs here are complete graphs with one edge missing and induced subgraphs with exactly one edge. The (4,6) and (4,7) Ramsey numbers for these types of subgraphs is computed. The method used is an exhaustive search, with many shortcuts employed to reduce computation time
