{"schema":"vela.problem-packet.v0.1","problem":562,"statement":"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 $r\\geq 3$,\\[\\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 $r-1$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}