Hypergraphs without non-trivial intersecting subgraphs

A hypergraph F is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of F . Mubayi and Verstraëte showed that for every k ≥ d + 1 ≥ 3 and n ≥ ( d + 1) k / d every k -graph H on n vertices without a non-trivial intersecting subgraph of size d + 1 contains at most (cid:2) n − 1 k − 1 (cid:3) edges. They conjectured that the same conclusion holds for all d ≥ k ≥ 4 and sufficiently large n . We confirm their conjecture by proving a stronger statement. They also conjectured that for m ≥ 4 and sufficiently large n the maximum size of a 3-graph on n vertices without a non-trivial intersecting subgraph of size 3 m + 1 is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.