Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.Prove that, for $r\ge 3$,$${\log }_{r-1}{R}_r(n){\asymp }_{r}n,$$where ${\log }_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like$$2^{2^{\cdots n}}$$where the tower of exponentials has height $r-1$?