Transversals in 4-uniform linear hypergraphs

Let H be a hypergraph of order n_H= |V(/H)| and size m_H= |E(H)|. The transversal number τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A k-uniform hypergraph has all edges of size k. For k ≥ 2, let L_kdenote the class of k-uniform linear hypergraphs. We consider the problem of determining the best possible constants q_k(which depends only on k) such that τ(H) < q_k{n_H+m_H) for all H ∈ L_k. It is known that q₂= 1/3 and q₃= 1/4. In this paper we show that q₄= 1/5, which is better than for non-linear hypergraphs. Using the affine plane AG(2,4) of order 4, we show there are a large number of densities of hypergraphs H ∈ L₄such that τ(H) ≈ 1/5{n_H+ m_H).