Uniform hypergraph Ramsey number ($R^{(r)}(k)$)

The diagonal $r$-uniform hypergraph Ramsey number $R^{(r)}(k)$ is the least $N$ such that every red-blue colouring of the $r$-element subsets of an $N$-element set has a monochromatic $k$-element set.

Let $K_4^{(3)-}$ be the three-uniform hypergraph with three of the four possible edges on four vertices. Every $K_4^{(3)-}$-free three-uniform hypergraph on $N$ vertices has an independent set of size at leastEquivalently, $r(K_4^{(3)-},K_k^{(3)})\le k^{Ck}$.