Upper Bound Of The Ramsey Number Of Studies

Since 1928 Ramsey put forward the famous Ramsey theorem,which have been aroused widely research in Ramsey problem.In these problems, Ramsey numbers research is very important, but very slow progress have been achieved in this study. At present, we just acquired several Ramsey values with various methods.So the Ramsey numbers always the famous difficult problem in combinatorial mathematics,discrete mathematics, graph theory and computer. Everybody seeks to the available methods to acquire the Ramsey numbers, but still does not find a reasonable way to resolve the problem.In this paper, we use three methods to resolve the Ramsey numbers. The first one is drawer principle.Using it,we proved the Ramsey numbers theorem and the theorem Rn(3)≤n(Rn-1(3)-1),the former was put forward by Erdos & Szekeres(1935),Greenwood & Gleason(1955). Meanwhile, we obtained the upper bound formula of two sorts of Ramsey number:Rn-1(k; k+1)≤n(Rn (k)-1)+2 and Rn-1(k;l+1)≤n(Rn-1(k;l)-1)+2.The second method is extension of the sum-free sets of integers, which is used to resolve the values of the Ramsey numbers.The third method is cyclic graph, using it, we study the upper and lower bounds for the Ramsey numbers.Firstly, we put forward a new computational method for lower bound on. And then we obtain the upper bound by this method.Finally,we use them to prove the classical three-color Ramsey number R(3,3,4)=30.