Smallest Transversals of Small \(3\)-graphs

A smallest transversal of a \(k\)-graph (or \(k\)-uniform hypergraph) is any smallest set of vertices that intersects all edges. We investigate smallest transversals of small (up to ten vertex) \(3\)-graphs. In particular, we show how large the smallest transversal of small \(3\)-graphs can be as a function of the number of edges and vertices. Also, we identify all \(3\)-graphs with up to nine vertices that have largest smallest transversals. This work is related to a problem of Turán, and to the covering problem. In particular, extremal \(3\)-graphs correspond to covering designs with blocks of size \(n-3\).