  The list Ramsey number $R_{\ell}(H,k)$, recently introduced by Alon, Buci\'c,
Kalvari, Kuperwasser, and Szab\'o, is a list-coloring variant of the classical
Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that
is not $r$-partite and the number of colors $k$ goes to infinity,
$e^{\Omega(\sqrt{k})} \le R_{\ell} (H,k) \le e^{O(k)}$. We prove that
$R_{\ell}(H,k) = e^{\Theta(k)}$ if and only if $H$ is not $r$-partite.
