Lines in hypergraphs (Q2439836)

The line \(uv\) in a \(3\)-uniform hypergraph \(H\) is a set of vertices of \(H\) consisting of \(u,v\) and all \(w\) such that \(\{u,v,w\}\) is a hyperedge in \(H.\) The main result of the paper states that in a \(3\)-uniform hypergraph \(H\) with no four vertices inducing two hyperedges and such that the set of vertices is not a line, the total number of lines is at least the number of vertices. This theorem constitutes another generalization of the De Bruijn-Erdős theorem on the number of hyperedges in hypergraphs where every two vertices belong to precisely one hyperedge.