Lower bounds on Tuza constants for transversals in linear uniform hypergraphs

Let H be a hypergraph of order n_H=|V(H)| and size m_H=|E(H)|. The transversal number τ(H) of 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. Let L_k be the class of k-uniform linear hypergraphs. In this paper we study the problem of determining or estimating 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. We establish lower bounds on q_k for all k≥2. For all k≥2 and for all 0<a<1, we show that q_k≥a(1−a)^k∕((1−a)^k+aln(e∕a)), where here ln denotes the natural logarithm to the base e.