Improved bounds for coloring locally sparse hypergraphs

We show that, for every $k\ge 2$, every $k$-uniform hypergaph of degree $\mathrm{\Delta }$ and girth at least $5$ is efficiently $(1+o(1))(k-1)(\mathrm{\Delta }\mathrm{/}\ln \mathrm{\Delta }{)}^{1\mathrm{/}(k-1)}$-list colorable. As an application (and to the best of our knowledge) we obtain the currently best algorithm for list-coloring random hypergraphs of bounded average degree.