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

OPEN - $500

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 $3$-uniform hypergraph on $n$ vertices.

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