Partial Ovoids and Spreads in Generalized Quadrangles, and Related Combinatorial Structures

In this paper we overview what is known about partial ovoids and spreads of finite (classical) generalized quadrangles. In the first, respectively the second, part of the paper we will be mostly concerned with small, respectively large, maximal partial ovoids and spreads. Also connections with other interesting objects in finite geometry will be explained. Among the new results are new bounds on the smallest and largest maximal partial spreads of Q(5, q) and H(3, q2), an improvement on a recent bound for small maximal partial ovoids of W (q3), and a new bound on small maximal partial spreads of H(4, q2). We also classify transitive maximal partial ovoids of size (q2 − 1) of Q(4, q), and discuss examples for small q. We introduce a theory for spreads and ovoids of affine generalized quadrangles, and apply earlier obtained results to this theory. Finally, we obtain new results in the study of maximal (s+1)×(s−1)-grids in thick generalized quadrangles of order (s, t), and connect these objects to certain maximal partial spreads. Acknowledgement. Both authors are Postdoctoral Fellow of the Fund for Scientific Research Flanders (FWO-Vlaanderen).