Dual Erdős–Faber–Lovász conjecture on chromatic index

A linear hypergraph is a hypergraph in which every pair of edges intersects in at most one vertex. The chromatic index of a hypergraph is the minimum number of colors needed to color its edges so that intersecting edges receive different colors.

Dual Erdős–Faber–Lovász conjecture. Any linear hypergraph on $n$ vertices has chromatic index at most $n$.

This is the dual formulation of the Erdős–Faber–Lovász coloring problem, obtained by interchanging vertices and edges. The conjecture is still widely open, although it is known for various hypergraph families and asymptotic and fractional analogues have been established.