Mathematisches Institut, Arndtstr. 2, D-35392 Giessen, Germany e-mail: Klaus.Metsch@math.uni-giessen.de University of Gent, ZWC, Galglaan 2, 9000 Gent, Belgium e-mail: ls@cage.rug.ac.be, http:/$\!$/cage.rug.ac.be/$\sim$ls
Abstract: We show that the three smallest minimal point sets of PG$(4,q)$, $q$ square, $q> 9$, that meet all planes are the set of points of a plane, the set of points in a Baer cone and the set of points in a Baer subgeometry PG$(4,\sqrt q)$. This implies that PG$(4,\sqrt q)$ is the unique smallest example of a set of points of PG$(4,q)$ that meets every plane and contains no line. It also implies that PG$(4,\sqrt q)$ is the unique smallest minimal set of points of PG$(4,q)$ that meets all planes and generates PG$(4,q)$.
Full text of the article: