On conjectures of Berge and Chvátal (Q1313827)

A hypergraph \(H\) is linear if any two of its edges share at most one vertex. The hereditary closure \(H^ \wedge\) of \(H\) is the collection of all nonempty subsets of the edges of \(H\). \(\Delta(H)\) is the maximum number of edges of \(H\) incident to a given vertex; \(\Delta_ 0(H)\) is the maximum number of pairwise intersecting edges in \(H\); \(q(H)\) is the chromatic index of \(H\). Conjecture 1 (C. Berge, 1990): If \(H\) is linear, then \(\Delta (H^ \wedge)=q(H^ \wedge)\). Conjecture 2 (V. Chvátal, 1974): If \(H=H^ \wedge\), then \(\Delta (H)=\Delta_ 0(H)\). The authors prove Conjecture 1 for all resolvable Steiner systems \(H\) of type \(S(2,k,n)\) and Conjecture 2 for every sufficiently large BIBD if the block size and the number of blocks containing a given pair of points are fixed.