Improved bounds for the chromatic index of $k$-uniform hypergraphs

In 1997, Alon and Kim conjectured that if $H$ is a $k$-uniform $t$-simple hypergraph with maximum degree $D$ sufficiently large, then the chromatic index $\chi'(H)$ is upper bounded by $(t-1+1/t+\varepsilon)D$. Using probabilistic techniques and a nibble coloring method, we prove a general coloring theorem stating that a $k$-uniform $t$-simple hypergraph $H$ with large maximum degree $D$ satisfies $$\chi'(H) \le (b+\varepsilon)kD,$$ where $b$ is a particular parameter derived from local structural information about $H$. We use structural techniques to prove sharp upper bounds on $b$ in the 3-uniform 2-simple, and 3-uniform 3-simple cases. In particular, we deduce as a corollary that for sufficiently large $D$, every 3-uniform 2-simple and 3-simple hypergraph of maximum degree at most $D$ has chromatic index at most $2.3581D$ and $2.6791D$, respectively.