A note on the Tuza constant c_k for small k

For a hypergraph H, the transversal is a subset of vertices whose intersection with every edge is nonempty. The cardinality of a minimum transversal is the transversal number of H, denoted by τ(H). The Tuza constant c_k is defined as sup⁡τ(H)/(m+n), where H ranges over all k-uniform hypergraphs, with m and n being the number of edges and vertices, respectively. We give an upper bound and a lower bound on c_k. The upper bound improves the known ones for k≥7, and the lower bound improves the known ones for k∈{7,8,10,11,13,14,17}.