Blowup Ramsey numbers

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 $G(n,p) \stackrel{r}{\rightarrow} H$.

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