Hypergraph Ramsey numbers with quasipolynomial growth rate

For a 3-uniform hypergraph (3-graph) $F$, let $r(F,n)$ be the smallest $N$ such that any $N$-vertex $F$-free 3-graph has an independent set of size $n$. We construct a $3$-graph $H_2$ with six vertices and five edges such that $r(H_2,n)=n^{\Theta(\log n)}$, and a more general family of $3$-graphs $F$ for which $r(F,n)=n^{\log^{\Theta(1)}(n)}$. These are the first examples of such Ramsey number known to be neither polynomial nor exponential.