A note on hypergraphs without non-trivial intersecting subgraphs

Abstract

A hypergraph F is non-trivial intersecting if every two 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)n/d every k-graph H on n vertices without a non-trivial intersecting subgraph of size d + 1 contains at most n−1 k−1 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 3m + 1 is achieved by certain Steiner systems. We give a construction with more edges showing that their conjecture is not true in general.