OPEN
This is open, and cannot be resolved with a finite computation.
- $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\[R_3(n) \geq 2^{2^{cn}}?\]
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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.
This problem is
#37 in Ramsey Theory in the graphs problem collection.
View the LaTeX source
This page was last edited 18 January 2026. View history
External data from
the database - you can help update this
Formalised statement?
Yes
Related OEIS sequences:
Possible
|
Likes this problem
|
Dogmachine
|
|
Interested in collaborating
|
None
|
|
Currently working on this problem
|
None
|
|
This problem looks difficult
|
None
|
|
This problem looks tractable
|
None
|
|
The results on this problem could be formalisable
|
None
|
|
I am working on formalising the results on this problem
|
None
|
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #564, https://www.erdosproblems.com/564, accessed 2026-04-11
