  For a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $G$ (or a
Berge-$G$ in short), if there is a bijection $f : E(G) \rightarrow
E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We
denote the family of $r$-uniform hypergraphs that are Berge copies of $G$ by
$B^rG$.
  For families of $r$-uniform hypergraphs $\mathbf{H}$ and $\mathbf{H}'$, we
denote by $R(\mathbf{H},\mathbf{H}')$ the smallest number $n$ such that in any
blue-red coloring of $\mathcal{K}_n^r$ (the complete $r$-uniform hypergraph on
$n$ vertices) there is a monochromatic blue copy of a hypergraph in
$\mathbf{H}$ or a monochromatic red copy of a hypergraph in $\mathbf{H}'$.
$R^c(\mathbf{H})$ denotes the smallest number $n$ such that in any coloring of
the hyperedges of $\mathcal{K}_n^r$ with $c$ colors, there is a monochromatic
copy of a hypergraph in $\mathbf{H}$.
  In this paper we initiate the general study of the Ramsey problem for Berge
hypergraphs, and show that if $r> 2c$, then $R^c(B^rK_n)=n$. In the case $r =
2c$, we show that $R^c(B^rK_n)=n+1$, and if $G$ is a non-complete 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 $R^c(B^rK_n)$. Moreover, we also determine the
exact value of $R(B^3T_1,B^3T_2)$ for every pair of trees $T_1$ and $T_2$.
