Problem 564

Erdős Problem #564: declared status 'open'. Formalized: yes. 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$$R_3(n) \geq 2^{2^{cn}}?$$ Current best: Erd\H{o}s, Hajnal, M\'{a}t\'{e}, and Rado [EHMR84] have proved a doubly exponential lower bound for the corresponding problem with $4$ colours. Prize: $500. Tags: graph theory, hypergraphs, ramsey theory.