LetH be an arbitrary r-uniform hypergraph. The q-color Ramsey number Rr(H; q) is the minimum integer n such that there is a monochromatic copy of H in every q-edge-coloring of Krn , complete r-uniform hypergraph of order n. The existence of such a positive integer is guaranteed by Ramsey’s ex(n;H). For a graph G, we say that a hypergraph H is a Berge-G if there is a bijection ? : E(G) ! E(H) such that e amp;#??; ?(e) for every e ? E(G). Determining the Ramsey numbers of r-uniform t-tight Berge- cycles and the Tur?n numbers of complete ?-uniform Berge-hypergraphs are two problems investigated in this thesis. Gy?rf?s, Lehel, S?rk?zy and Schelp conjectured that for r amp;#??; ? and sufficiently large n, every (r???)-edge coloring of complete hypergraphKrn contains a monochromatic Hamiltonian Berge-cycle. They proved their conjecture for the first case r = ? and a weaker form of conjecture, which indicates that the statement of this conjecture is true for sufficiently large n with ?r??? ? ? colors instead of r??? colors. An asymptotic form of conjecture was proved for every r, using the method of Regularity Lemma, by Gy?rf?s, S?rk?zy and Szemerédi. In this thesis, we show that this conjecture is true for the case r = ?, the first open case. Also we prove a weaker form of conjecture by using r ??? colors instead of r ??? colors. Some new results are also given for the Ramsey numbers of r-uniform t-tight Berge- cycles. Considering the problem of finding the Tur?n numbers of Berge-hypergrahs, in this thesis, we determine the exact value of the Tur?n numbers of complete ?-uniform Berge-hypergraphs and present their extremal construction. کلیدواژه فارسی
