The list coloring number of uncrowded hypergraphs

We prove that for every fixed integer $r\geq 2$ and every $\varepsilon>0$, every sufficiently large finite uncrowded $(r+1)$-uniform hypergraph of maximum degree $\Delta$ has list chromatic number at most \[
(1+\varepsilon)\left(\frac{r\Delta}{\log\Delta}\right)^{1/r}. \] The proof is a semi-random list-coloring nibble carried out directly on the original hypergraph. We encode the remaining coloring problem by active edge-color constraints and control all residual sizes through a binomial degree bound. After the nibble reaches a sparse terminal state, the coloring is completed by a Rosenfeld-style counting argument.