Determining Triangle Ramsey Numbers

The problem of determining the Ramsey numbers comes from sociology. Sociologists wondered whether the properties that they observe in certain social graphs are random or somehow hold in all possible cases. Besides, in computation theory researchers are interested in finding huge cliques in graphs. The Ramsey graph $(G, H)$ is a graph that does not contain a sub-graph isomorphic to $G$ and its complement does not contain a sub-graph isomorphic to $H$. This thesis is about $(K_3, B_k)$ Ramsey graphs, where $B_k$ is a clique of size $k$ without 2 edges, which do not share any vertices. In this work are computed all Ramsey graphs $(K_3, B_k)$ and Ramsey numbers R $(K_3, B_k)$, where $k$ is not bigger than 8. We believe, that there are described all necessary theoretical properties and algorithms for computing Ramsey graphs and Ramsey numbers for k equals 9 and we hope, that this technique works for $k$ equals 10 too, so this will solve one of 7 open cases of Ramsey numbers of order 10.