Multicolor list Ramsey numbers grow exponentially

The list Ramsey number R-l(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(root k)) <= R-l(H,k) <= e(O(k)). We prove that R-l(H,k) = e(Theta(k)) if and only if H is not r-partite.