In general, our project was successful in achieving its objectives. While we still have many more ideas to even further improve our program and computational model, we can still draw many valuable conclusions form the current program and its various performance enhancements. These conclusions include the following:
- A computer and logic-based approach to calculating Ramsey numbers is possible, reaffirming previous research. Only after gathering research to understand the complex ideas behind calculating Ramsey numbers did we even start to apply computer-based thinking to calculating Ramsey numbers. This clearly paid off as we too were able to conclude that it is possible to calculate Ramsey numbers on the computer like the few, past researchers who have tried.
- Our basic algorithms that cycle through colorings, subgraphs, and edges is accurate as the program that we created with these algorithms produced accurate results. Generating the larger framework in which the calculate Ramsey numbers is a challenge in itself both conceptually and computationally, and having solved it, we have created a solution available for future researchers.
- Furthermore, the algorithms that we developed to reduce the number of colorings tested are both accurate created significant time savings. These algorithms improved upon our original, somewhat slow program and created a program that succeeded in reducing the amount of time for calculations. In some cases, the time is reduced by 80% and even more time could be removed in higher sizes. So ultimately, the ideas and algorithms behind our program have all worked to effectively reduce the processing time over a “brute force”-type method.
- Even though the time savings as a result of our innovative algorithms are significant, the are not nearly enough to overcome the significant increases in required processing power for larger and larger graphs. In running the program, we found that processing power skyrocketed when we even added one node to a graph. Nevertheless, the algorithms we developed could be used as part of a larger project to calculate Ramsey numbers on a supercomputer, and with other innovative algorithms, possibly simplify the problem to a manageable size for larger graph sizes.
- As this program accurately calculates Ramsey numbers and does so with a number of time savers, the program fulfill the original intentions and specifications of the project.
These findings, especially the ones relating to successful algorithms, are based on the data and analysis from numerous runs with different inputs as well as visual proof provided by the second, graphing program. This program showed coloring with no monochromatic subgraphs and allowed the user to self-verify the first, computational program’s findings.