  For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an
$R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs
to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of
$\mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a
hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if
there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e
\in E(G)$, $e \subseteq f(e)$. In this note, we define a new type of Ramsey
number, namely the \emph{cover Ramsey number}, denoted as $\hat{R}^R(BG_1,
BG_2)$, as the smallest integer $n_0$ such that for every covering $R$-uniform
hypergraph $\mathcal{H}$ on $n \geq n_0$ vertices and every $2$-edge-coloring
(blue and red) of $\mathcal{H}$ , there is either a blue Berge-$G_1$ or a red
Berge-$G_2$ subhypergraph. We show that for every $k\geq 2$, there exists some
$c_k$ such that for any finite graphs $G_1$ and $G_2$, $R(G_1, G_2) \leq
\hat{R}^{[k]}(BG_1, BG_2) \leq c_k \cdot R(G_1, G_2)^3$. Moreover, we show that
for each positive integer $d$ and $k$, there exists a constant $c = c(d,k)$
such that if $G$ is a graph on $n$ vertices with maximum degree at most $d$,
then $\hat{R}^{[k]}(BG,BG) \leq cn$.
