  A fundamental problem in Ramsey theory is to determine the growth rate in
terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform
hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We
study this problem, proving two main results. First, we show that for a broad
class of $H$, including links of odd cycles and tight cycles of length not
divisible by three, $r(H, K_n^{(3)}) \ge 2^{\Omega_H(n \log n)}$. This
significantly generalizes and simplifies an earlier construction of Fox and He
which handled the case of links of odd cycles and is sharp both in this case
and for all but finitely many tight cycles of length not divisible by three.
Second, disproving a folklore conjecture in the area, we show that there exists
a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$.
This provides the first example of a separation between $r(H,K_n^{(3)})$ and
$r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when
$H$ is linear.
