An ovoid of a finite 3-dimensional projective space $PG(3, q)$ is a set of $q^2 + 1$ points such that no three are on a common line. Besides the case $q=2$, the bound $q^2+1$ is the maximum size of a set of points with no three collinear. Ovoids are central to finite geometry, and two examples of their significance are the constructions of inversive planes and generalised quadrangles from ovoids. Peter Dembowski (1963) proved that every inversive plane of even order arises from an ovoid of $PG(3,q)$, $q$ even, and Jacques Tits (see Dembowksi’s book) gave a construction of a generalised quadrangle of order $(q,q^2)$ from an ovoid. Here is a simple example of an ovoid: Continue reading “Open problems in “Finite Geometry”: Ovoids of projective spaces”