Erdős Problem #562

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.