Multicolor list Ramsey numbers grow exponentially

The list Ramsey number $R_{ell}(H,k)$, recently introduced by Alon, Bucic, Kalvari, Kuperwasser, and Szabo, 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.