Dense uniform hypergraphs have high list chromatic number

The first author showed that the list chromatic number of every graph with average degree $d$ is at least $(0.5?o\left(1\right)){log}_{2}d$. We prove that for $r\ge 3$, every $r$-uniform hypergraph in which at least half of the $(r?1)$-vertex subsets are contained in at least $d$ edges has list chromatic number at least $\frac{lnd}{100r^3}$. When $r$ is fixed, this is sharp up to a constant factor.