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.
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