Berge–Füredi conjecture for linear hypergraphs

Let $(V,E)$ be a linear, loopless hypergraph: for distinct hyperedges $V_i,V_j\in E$, $|V_i\cap V_j|\leq 1$, and every hyperedge has more than one vertex. Let $q(V,E)$ denote its chromatic index, and let $[(V,E)]_2$ be its 2-section, obtained by replacing every hyperedge with a complete graph on its vertices. Write $\Delta([(V,E)]_2)$ for the maximum degree of this graph.

Berge–Füredi conjecture. A linear (loopless) hypergraph $(V,E)$ satisfies

$$q(V,E) \leq \Delta([(V,E)]_2)+1.$$

This is the hypergraph analogue of Vizing's upper bound for the chromatic index of a graph. The paper presents results suggesting an approach to the conjecture and gives sufficient conditions for it to hold, including conditions involving the Helly property; its general validity remains open.