Ramsey Numbers: Improving the Bounds of R(5,5)

Ramsey number R(m,n) = r is the smallest integer r such that a graph of r vertices has either a complete subgraph (clique) of size m or its complement has a complete subgraph of size n. Currently, the exact value of R(5,5) is unknown, however the best known lower and upper bounds are 43 <= R(5,5) <= 49. In this paper we will discuss a method that we use to construct a better lower bound, namely, by way of genetic programming (GP)/ genetic algorithm (GA). This method involves standard genetic algorithms mutations and crossovers as recombination techniques as well as using algorithms and heuristics to find maximum clique of a graph. We implemented this method on a particular genetic algorithm software called Sutherland. Results will include the “best” graphs found using this technique over multiple runs, statistical data as to the likelihood of increasing the current best-known lower bound of R(5,5) if not strictly the lower bound.