  We study a generalisation of the bipartite Ramsey numbers to blowups of
graphs. For a graph $G$, denote the $t$-blowup of $G$ by $G[t]$. We say that
$G$ is $r$-Ramsey for $H$, and write $G \stackrel{r}{\rightarrow} H$, if every
$r$-colouring of the edges of $G$ has a monochromatic copy of $H$. We show that
if $G \stackrel{r}{\rightarrow} H$, then for all $t$, there exists $n$ such
that $G[n] \stackrel{r}{\rightarrow} H[t]$. In fact, we provide exponential
lower and upper bounds for the minimum $n$ with $G[n] \stackrel{r}{\rightarrow}
H[t]$, and conjecture an upper bound of the form $c^t$, where $c$ depends on
$H$ and $r$, but not on $G$. We also show that this conjecture holds for
$G(n,p)$ with high probability, above the threshold for the event $G(n,p)
\stackrel{r}{\rightarrow} H$.
