Figure 1: The three input prompts for the calculation program. If this program were to be run on a supercomputer, these prompts could be changed to console window prompts. These would be the inputs for a R(3, 3) with a guess of 5.

Figure 2: The error message that comes up if the user of the computational program puts in an invalid input. All inputs into that program have error catchers to ensure that only valid inputs are used in the program.

Figure 3: The console output window for the inputs in Figure 1. As the table in Appendix C states, R(3, 3,) is > the guess, 5. The Edge Sequence Key, displayed because there is a graphing with no monochromatic colorings, is 5X3X3X0100111100.

Figure 4: The input window for the second, graphing program. This input asks for the Edge Sequence Key from the calculations program. The one entered is the from the output in Figure 3.

Figure 5: The graph printed from the Edge Sequence Key from Figure 4. Note that as the window’s text states, there are no monochromatic k 3‘s in either colors.

Figure 6: The console output for a R(3, 3) with a guess of 6. The program does not find a graph without monochromatic colorings, so the Ramsey number is less than or equal to the guess. As Figures 1 and 3-5 showed that R(3, 3) is > 5, and in this figure the outputs shows R(3, 3) ≤ the guess 6, so the answer must be six, which it is.

Figure 7: The console output with Edge Sequence Key. The inputs were an R(3, 4) with guess 7. The total processing power time with all enhancement algorithms was 9 seconds compared to an unimproved algorithm that ran for at least 13.25 seconds. The Edge Sequence Key is: 7X4X3X101001100110011111000.

Figure 8: The graphing for the output from Figure 7. In this case, the color of the edges is important to take note of because, here, there may be as many blue k 3 as we need because its subgraph size is 4. There may be no red k 3’s, however, as 3 is the size of the subgraph assigned to red.