In this thesis, we study the diagonal Ramsey numbers of particular the edges of the complete graph K n with distinct colors { 1 , 2 , ..., k } , there is a monochro- matic copy of G i in color i , for some 1 ? i ? k . A problem on Ramsey numbers of general graphs was posed in 1983 by Erd?os. Erd?os conjectured that there is a constant c such ? that r ( G ) ? 2 c m for every graph G with m edges and no isolated vertices. In 2011, Sudakove proved this conjecture. Here we present the Sudukov’s result and then give an extension of his result. Determining Ramsey number of sparse graphs is also an interesting roblem in graph Ramsey theory. We say that a family of graphs G is a Ramsey linear family if there is a constant c = c ( G ) 0 such that r ( G ) ? cn for every graph G ? G of order n . urr and Erd?os conjectured: 1. The family of graphs having maximum degree at most ? is Ramsey linear. 2. The family of d -degenerate graphs is Ramsey linear. The first conjecture was proved y Chvatal et al. in 1983 but the econd one is still open. For an k -uniform hypergraph H , the Ramsey umber r k ( H ) is defined as similar to this umber for graphs. Kostochka and R¨odl gave a counterexample howing that family of k -uniform and d -degenerate hypergraph is not Ramsey linear. agle et al. and Cooley et al., using the zemer´edi’s regularity lemma, how that the family of k - uniform hypergraphs with bounded maximum degreee is Ramsey linear. In 2009, Conlon, Fox and Sudakov used a probabilistic argument known as dependent random choice to give another proof for the above fact. Here we present the results obtained by Conlon et al. The multicolor Ramsey number r ( P n 1 , ..., P n t ) is not known for t ? 3. Here we deter- mine the Ramsey number r ( P 3 , P n , P m ) for all m ? n .
