For given k-uniform hypergraphs G and H, the Ramsey number R(G,H) is defined to be the smallest integer N so that in every red-blue coloring of the edges of the complete k-uniform hypergraph , there is either a red copy of G or a blue copy of H. Haxell et al. proved that the 2-color Ramsey number of 3-uniform loose cycles on 2n vertices is asymptotically . This result was extended by Gyarfas, Sarkozy and Szemeredi to k-uniform loose cycles. In this thesis, we provide the exact values of 2-color Ramsey numbers involving loose paths and cycles in 3 -uniform hypergraphs. These give a positive answer to a question of Gyarfas and Raeisi. In the sequel, we conjecture that for every n? m? 3 and k? 3, [ ]. We show that how the truth of the last equality of the conjecture leads to the truth of the hole of conjecture. We also demonstrate that for a fixed m? 3 and k? 4, if the conjecture holds for $m\\leq n\\leq 2m,$ then it holds for all n? m. Subsequently, the conjecture follows for m=3. In addition, we study the Ramsey number of loose cycles in k -uniform hypergraphs when k= 4,5 and k ? 8 .
