In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces \({{\,\textrm{PG}\,}}(n, q)\) such that every plane is incident with at least one line of the set. This is the first main open problem concerning the minimum size of (s, t)-blocking sets in \({{\,\textrm{PG}\,}}(n,q)\) , where we set \(s=2\) and \(t=1\) . In \({{\,\textrm{PG}\,}}(n,q)\) , an (s, t)-blocking set refers to a set of t-spaces such that each s-space is incident with at least one chosen t-space. This is a notoriously difficult problem, as it is equivalent to determining the size of certain q-Turán designs and q-covering designs. We present an improvement on the upper bounds of Etzion and of Metsch via a refined scheme for a recursive construction, which in fact enables improvement in the general case as well.
