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 + a ln ( e ∕ a ) ) , where here ln denotes the natural logarithm to the base  e .