For graphs $G$ and $H$, we say $G \stackrel{r}{\to} H$ if every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. The blowup Ramsey number $B(G \stackrel{r}{\to} H;t)$ is the minimum $n$ such that $G[n] \stackrel{r}{\to} H[t]$. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given $G$, $H$ and $r$ such that $G \stackrel{r}{\to} H$, there exist constants $a=a(G,H,r)$ and $b=b(H,r)$ such that for all $t \in \mathbb{N}$, $B(G \stackrel{r}{\to} H;t) \leq ab^t$. They conjectured that there exist some graphs $H$ for which the constant $a$ depending on $G$ is necessary. We prove this conjecture by showing that the statement is true when $H$ is a $3$-chromatically connected, which includes all cliques on $3$ or more vertices. We are also able to show perhaps surprisingly that for any forest $F$ there is $f(F,t)$ such that for any $G \stackrel{r}{\to} H$, $B(G \stackrel{r}{\to} H;t)\leq f(F,t)$ i.e. the function does not depend on the ground graph $G$. This is joint work with Robert Hancock.
