Problem 564 — Visual explainer (undergrad)
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Problem statement
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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\[R_3(n) \geq 2^{2^{cn}}?\]

Picture
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Graph picture:

o---o
|\  |
| \ |
o---o    K4 is the complete graph on 4 vertices (all connections)

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
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Area hint: infinite graph theory / Ramsey-type decompositions.

Benchmarks / known results (from comments)
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- A problem of Erd\H{o}s, Hajnal, and Rado, 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 have proved a doubly exponential
  lower bound for the corresponding problem with $4$ colours.