  Recently, Souza introduced blowup Ramsey numbers as a generalization of
bipartite Ramsey numbers. For graphs $G$ and $H$, say
$G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a
monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the
blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such
that $G[n] \overset{r}{\longrightarrow} H[t]$. Souza proved upper and lower
bounds on $n$ that are exponential in $t$, and conjectured that the exponential
constant does not depend on $G$. We prove that the dependence on $G$ in the
exponential constant is indeed unnecessary, but conjecture that some dependence
on $G$ is unavoidable.
  An important step in both Souza's proof and ours is a theorem of Nikiforov,
which says that if a graph contains a constant fraction of the possible copies
of $H$, then it contains a blowup of $H$ of logarithmic size. We also provide a
new proof of this theorem with a better quantitative dependence.
