A special case of [562]. A problem of Erd\H{o}s, Hajnal, and Rado \cite{EHR65}, who prove the bounds\[2^{cn^2}< R_3(n)< 2^{2^{n}}\]for some constant $c>0$.

Erd\H{o}s, Hajnal, M\'{a}t\'{e}, and Rado \cite{EHMR84} have proved a doubly exponential lower bound for the corresponding problem with $4$ colours.

This problem is #37 in Ramsey Theory in the graphs problem collection.


References


[EHMR84] Erd\H{o}s, Paul and Hajnal, Andr\'{a}s and M\'{a}t\'{e}, Attila and Rado, Richard, Combinatorial set theory: partition relations for cardinals. (1984), 347.

[EHR65] Erd\H{o}s, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.