Ovoids in infinite incidence structures (Q1842008)

Using simple transfinite induction, it is shown that a large class of infinite incidence structures (including polar spaces and most dual polar spaces, and generalized \(n\)-gons with \(n \geq 4\) and \(s= t\)) have the property that the point set can be partitioned into ovoids. As a result, there seems little hope of describing universal embeddings of these geometries in terms of hyperplanes. On the other hand, combined with a construction of Kantor, the result gives rise to large numbers of flat geometries with diagrams such as \(C_ 2 \cdot C_ 2\).