Hypergraph Ramsey numbers of cliques versus stars

Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_4^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, $$2^{c{\log }^{2}n}\le r(K_4^{(3)},S_n^{(3)})\le 2^{c^{\mathrm{\prime }}{n}^{2\mathrm{/}3}\log n}$$ for some positive constants $c$ and $c^{\mathrm{\prime }}$. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum $N$ such that any $2$-edge-coloring of the Cartesian product $K_N\mathrm{\square }{K}_N$ contains either a red rectangle or a blue $K_n$?