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For a hypergraph $H$, the transversal is a subset of vertices whoseintersection with every edge is nonempty. The cardinality of a minimumtransversal is the transversal number of $H$, denoted by $\tau(H)$. The Tuzaconstant $c_k$ is defined as $\sup{\tau(H)/ (m+n)}$, where $H$ ranges over all$k$-uniform hypergraphs, with $m$ and $n$ being the number of edges andvertices, respectively. We give an upper bound and a lower bound on $c_k$. Theupper bound improves the known ones for $k\geq 7$, and the lower bound improvesthe known ones for $k\in\{7, 8, 10, 11, 13, 14, 17\}$.

A note on the Tuza constant $c_k$ for small $k$