Blocking set

Remarks

Some authors include "contains no line" in the definition of a blocking set.

Bruen's bound: a nontrivial blocking set in a plane of order $n$ has at least $n + \sqrt{n} + 1$ points, with equality exactly for the point set of a Baer subplane.

Example: let $q$ be a prime power and view the plane $\mathrm{PG}(2, q)$ as a subplane of $\mathrm{PG}(2, q^2)$. Its $q^2 + q + 1$ points form a nontrivial blocking set of $\mathrm{PG}(2, q^2)$. Check: each point of the subplane lies on $q^2 + 1$ lines of the large plane, of which $q + 1$ are lines of the subplane. A line of the large plane through two points of the subplane is a line of the subplane, so the other $q^2 - q$ lines meet the subplane in one point. This gives $(q^2 + q + 1)(q^2 - q) = q^4 - q$ such lines, and with the $q^2 + q + 1$ lines of the subplane that is $q^4 + q^2 + 1$, all lines of $\mathrm{PG}(2, q^2)$. No line of the large plane lies inside the subplane, since it has $q^2 + 1 > q + 1$ points.

Non-example: the point set of a line $\ell$ meets every line, but it contains $\ell$, so it is a trivial blocking set.