Planar and affine spaces (Q1043647)

Three-dimensional affine spaces are characterized as the only linear spaces endowed with a set \(\Omega\) of proper subspaces having the following properties: (1) Every line contains a constant number \(n\), \(n>2\), of points. (2) Every triple of noncollinear points is contained in a unique member of \(\Omega\). (3) `Disjoint or coincide' is an equivalence relation in \(\Omega\), and every equivalence class covers all points. The case \(n=2\) (when the structure is a complete graph endowed with a set \(\Omega\) of proper complete subgraphs) is also considered. It turns out that besides the affine 3-space of order 2, only two small additional examples occur. Finally, the authors characterize all finite affine spaces of dimension at least 3 having lines of size at least 3.