Alon–Kim conjecture on the chromatic index of simple hypergraphs

Let $\mathcal{H}$ be a hypergraph, and let $\chi'(\mathcal{H})$ denote the smallest number of colors needed to color its edges so that intersecting edges receive distinct colors. The hypergraph is $k$-uniform if every edge has size $k$, $t$-simple if every two distinct edges intersect in at most $t$ vertices, and has maximum degree at most $D$ if every vertex belongs to at most $D$ edges.

Alon–Kim conjecture. For every $k\ge t\ge 1$ and $\varepsilon>0$, there exists $D_0$ such that, for every $D\ge D_0$, every $k$-uniform, $t$-simple hypergraph with maximum degree at most $D$ satisfies

$$\chi'(\mathcal{H})\le (t-1+1/t+\varepsilon)D.$$

Alon and Kim showed that this bound would be asymptotically tight whenever there is a projective plane of order $t-1$. The cases $k=2$, $t=1$, and $t=2$ follow from classical graph or hypergraph edge-coloring theorems, but the conjecture remains open in general; this paper proves new bounds for several specific classes.