For any $g\ge 2$, if $n$ is sufficiently large and $\equiv 1,3(\mathrm{mod}6)$ then there exists a 3-uniform hypergraph on $n$ vertices such that every pair of vertices is contained in exactly one edge (i.e. the graph is a Steiner triple system) and for any $2\le j\le g$ any collection of $j$ edges contains at least $j+3$ vertices.