Füredi–Kahn–Seymour conjecture on chromatic indices of uniform hypergraphs

Let $\ell$ be a fixed positive integer. An $\ell$-uniform multihypergraph is a multihypergraph in which every edge contains exactly $\ell$ vertices, and let $\chi'(H)$ denote its chromatic index. For sufficiently large $\Delta$, suppose that $H$ has maximum degree at most $\Delta$.

Füredi–Kahn–Seymour conjecture. Then

$$\chi'(H)\leq \left(\ell-1+\frac{1}{\ell}\right)\Delta+o(\Delta).$$

This conjecture improves the trivial bound of roughly $\ell\Delta$. The paper notes that the relevant asymptotic bounds are sufficient for its simultaneous edge-colouring results; the conjecture itself is presented as an external conjecture.