We know that the (2-color) Ramsey-numbers for $3$-uniform hypergraphs are between roughly $2^{n^2}$ and $2^{2^n}$, and the situation is similar to $k$-uniform hypergraphs for every $k\ge 3$. (A recent paper.)

Are there any consequences of these Ramsey-number being close to either the above upper or the lower bound?

Here I don't mean the obvious consequence that we get about the Ramsey-numbers that come from the stepping-up lemma, but some possible solutions to conjectures/complexity results etc.


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