Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set ΩΩ of proper subspaces having the properties (1) every line contains a constant number of points, say nn, with n>2n>2; (2) every triple of noncollinear points is contained in a unique member of ΩΩ; (3) disjoint or coincide is an equivalence relation in ΩΩ with the additional property that every equivalence class covers all points. We also take a look at the case n=2n=2 (in which case we have a complete graph endowed with a set ΩΩ of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.