Erdős Problem 564

Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.

Is there some constant $c>0$ such that $$ R_3(n) \geq 2^{2^{cn}}? $$

Reference: erdosproblems.com/564

theorem Erdos564.erdos_564 : sorry ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, 2 ^ 2 ^ (c * n) ≤ ↑(Combinatorics.hypergraphRamsey 3 n)

Erdős Problem 564

Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.

Is there some constant $c>0$ such that $$ R_3(n) \geq 2^{2^{cn}}? $$