When are off-diagonal hypergraph Ramsey numbers polynomial?

A natural open problem in Ramsey theory is to determine those $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H, K_n^{(3)})$ grows polynomially with $n$. We make substantial progress on this question by showing that if $H$ is tightly connected or has at most two tight components, then $r(H, K_n^{(3)})$ grows polynomially if and only if $H$ is contained in an iterated blowup of an edge.