In [T. Szőnyi and Z. Weiner, “On some stability theorems in finite geometry”, manuscript] it is shown that if \(B\) is a set of points in \(\text{PG}(2,q)\) and \(P\) is a point not in \(B\) that is incident with precisely \(r\) lines which meet \(B\), then the total number of lines meeting \(B\) is at most \(1+rq+(| B| -r)(q+1-r)\). The proof of this fact is algebraic.
In the paper under review the author uses this result as a key ingredient in a very clever combinatorial induction argument to determine the smallest number of \(s\)-subspaces in \(\text{PG}(n,q)\) that miss a point set of given cardinality. In particular, it is shown that the number of lines in \(\text{PG}(3,q)\) missing a set of \(b\) points, with \(q+1 < b \leq q^2+q+1\), is at least \((q^2+q+1-b)q^2\), and moreover this minimum can be achieved only for a planar set of \(b\) points that meets every line in that plane.