= Solution
The blue hypergraph in the question is the three-edge hypergraph $K_4^{(3)-}$. The <Erdős-Hajnal bound for the three-edge hypergraph on four vertices> states that every $K_4^{(3)-}$-free three-uniform hypergraph on $N$ vertices has an independent set of size at least
$$
c\frac{\log N}{\log\log N}.
$$
Its proof exposes vertices successively and studies their link graphs. A blue triangle in a link is exactly a blue $K_4^{(3)-}$, so all links are triangle-free; applying the <Shearer independence bound for a triangle-free graph> in dyadic degree ranges either adds vertices to the independent set or leaves a reservoir whose logarithm decreases by only $O(\log\log N)$ per selected vertex. Iteration yields the displayed lower bound.
Now take $N=k^{Ck}$. For sufficiently large $C$,
$$
c\frac{\log N}{\log\log N}
=c\frac{Ck\log k}{\log(Ck\log k)}
\ge k.
$$
Thus, if there is no blue copy of $H$, the blue hypergraph has an independent $k$-set, which is a red $K_k^{(3)}$. Therefore
$$
\boxed{r(H,K_k^{(3)})\le k^{Ck}.}
$$
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