Ramsey problems for berge hypergraphs

For a graph G, a hypergraph scrH is a Berge copy of G (or a Berge-G in short) if there is a bijection f: E(G) rightarrow E(scrH ) such that for eache in E(G) we have esubseteq f(e). We denote the family of r-uniform hypergraphs that are Berge copies of G by B^rG. For families of r-uniform hypergraphs H and H^prime, we denote by R(H, H^prime ) the smallest number n such that in any red-blue coloring of the (hyper)edges ofscrK _n^r (the complete r-uniform hypergraph on n vertices) there is a monochromatic blue copy of a hypergraph in H or a monochromatic red copy of a hypergraph in H^\prime. R^c(H) denotes the smallest number n such that in any coloring of the hyperedges of scrK _n^r with c colors, there is a monochromatic copy of a hypergraph in H. In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if r > 2c, then R^c(B^rKn) = n. In the case r = 2c, we show that R^c(B^rKn) = n+ 1, and if G is a noncomplete graph on n vertices, then R^c(B^rG) = n, assuming n is large enough. In the case r < 2c we also obtain bounds on R^c(B^rKn). Moreover, we also determine the exact value of R(B³T₁, B³T₂) for every pair of trees T₁ and T₂,.