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 complete $3$-uniform hypergraph on $n$ vertices.
Is there some constant $c>0$ such that
\[R_3(n) \geq 2^{2^{cn}}?\]
A special case of
[562]. A problem of Erdős, Hajnal, and Rado
[EHR65], who prove the bounds
\[2^{cn^2}< R_3(n)< 2^{2^{n}}\]
for some constant $c>0$.
Erdős, Hajnal, Máté, and Rado [EHMR84] have proved a doubly exponential lower bound for the corresponding problem with $4$ colours.
See also the entry in the graphs problem collection.
