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import FormalConjecturesUtilErdős Problem 562
open Combinatorics Filter Realopen scoped Asymptotics
namespace Erdos562
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.
@[category research open, AMS 5]
theorem erdos_562 : answer(sorry) ↔
∀ r ≥ 3, (fun n ↦ log^[r - 1] (hypergraphRamsey r n)) =Θ[atTop] (fun n ↦ (n : ℝ)) := ⊢ True ↔ ∀ r ≥ 3, (fun n => log^[r - 1] ↑(hypergraphRamsey r n)) =Θ[atTop] fun n => ↑n
All goals completed! 🐙
end Erdos562