On an application of Hall's representatives theorem to a finite geometry problem (Q1112047)

The authors consider finite planar spaces, i.e. finite linear spaces together with a family of subspaces (``planes'') such that each non- collinear triple of points determines exactly one plane. They provide an elementary proof for the fact that the number of planes is at least as large as the number of points; they also describe the possible structure of planar spaces achieving equality. These results are derived by using P. Hall's marriage theorem. As the authors point out, similar results have been obtained in terms of lattice and matroid theory previously.