Using computational methods to increase Ramsey lower bounds

Abstract

Ramsey theory explores the conditions under which order must emerge in large structures, and Ramsey numbers quantify the threshold at which this order is inevitable. Despite significant effort, many Ramsey numbers remain unresolved, with only lower and upper bounds currently known. This project aims to advance the search for improved lower bounds through computational experimentation. Brute-force methods and genetic algorithms were implemented to search for counterexamples to R(5,5)=43 and R(4,6)=36, with the goal of disproving these as true Ramsey numbers. Despite extensive computation, no counterexamples were found, providing additional empirical support for these conjectured values. By combining literature review with algorithmic innovation, this project contributes to the computational toolkit available for extremal combinatorics and offers new insights into the structure of Ramsey graphs.