The list coloring number of uncrowded hypergraphs

We prove that for every fixed integer $r\ge 2$ and every $\epsilon >0$, every sufficiently large finite uncrowded $(r+1)$-uniform hypergraph of maximum degree $\mathrm{\Delta }$ has list chromatic number at most $$(1+\epsilon ){\left(\frac{r\mathrm{\Delta }}{\log \mathrm{\Delta }}\right)}^{1\mathrm{/}r}\mathrm{.}$$ 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.