Determining Triangle Ramsey Numbers

The Ramsey number R(m, n) is defined as the minimal number of vertices for which each graph contains either a clique of size m or an independent set of size n as an induced subgraph. For some pairs of m and n a particular Ramsey number is known. However, even for relatively small values there are only upper and lower estimates found so far. There are two approaches to finding the exact value of a Ramsey number: by exhaustive searches of all graphs of a given size and constructing mathematical proof. Exhaustive searches do not work well in practice: even the value R(5, 5) is as yet unknown, despite much research to speed up such searches. Proofs of such facts as R(1,n)=1 and R(2,n)=n for all n are trivial, but for triangle Ramsey numbers the exact values of R(3, n) for n > 9 are unknown. Despite the simplicity of the definition of a Ramsey number, most of the existing mathematical proofs are complicated and cannot be applied to new Ramsey numbers. An example of such a proof can be found in <<Some Graph Theoretic Results Associated with Ramsey's Theorem>>, where, along with a number of useful results in graph theory, there are estimations of multiple Ramsey numbers, including proofs of R(3, 6)=18 and R(3, 7)=23. In paper <<On the Ramsey number R (3, 6)>> the proof of R(3, 6) <= 18 is given. In my thesis I am showing a simplified version of this proof, prove R(3, 7) <= 24 and suggest approach to building a proof of R(3, 7) <= 23. Keywords: Graph Theory, Triangle Ramsey Numbers, Combinatorics