Transversals in \(4\)-uniform linear hypergraphs

Abstract

Let \( H \) be a hypergraph of order \( n_H = |V(H)| \) and size \( m_H = |E(H)| \). The transversal number \( \tau(H) \) 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 \( k \geq 2 \), let \( \mathcal{L}_k \) denote 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 \( \tau(H) \leq q_k(n_H + m_H) \) for all \( H \in \mathcal{L}_k \). It is known that \( q_2 = \frac{1}{3} \) and \( q_3 = \frac{1}{4} \). In this paper we show that \( q_4 = \frac{1}{5} \), 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 \( H \in \mathcal{L}_4 \) such that \( \tau(H) = \frac{1}{5} (n_H + m_H) \).