A lower bound on blocking semiovals
A set of points \(S\) in a projective plane \(\Pi\) which is at the same time a semioval and a blocking set is called blocking semioval, i.e., a blocking semioval is a set of points of \(\Pi\) satisfying the following properties: for every point \(P\in S\), there exists a unique line \(l\) of \(\Pi\) such that \(l\cap S=\{P\}\); every line of \(\Pi\) contains at least one point of \(S\), but is not entirely contained in \(S\). In this paper the author gives some lower bounds on the size of a blocking semioval in a finite projective plane. He proves that if \(\Pi\) is a projective plane of order \(q\) and \(S\) is a blocking semioval in \(\Pi\), then \(|S|\geq 2q+1\), and if \(q\geq 7\), then \(|S|\geq 2q+2\). If \(q=5\) or \(q=7\), these bounds are sharp. Also he constructs a family of blocking semiovals of size \(3q-4\) in the projective plane \(PG(2,q)\) with \(q>5\) odd.
scientific article; zbMATH DE number 1512032