Enumeration of points, lines, planes, etc

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points E in a projective plane determines at least |E| lines, unless all the points are contained in a line. The result was extended to higher dimensions by Motzkin and others, who showed that every set of points E in a projective space determines at least |E| hyperplanes, unless all the points are contained in a hyperplane. Let E be a spanning subset of an r-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of E, there are at least as many (r-p)-dimensional subspaces as there are p-dimensional subspaces, for every p at most 1\2r. This confirms the Ȝtop-heavy” conjecture by Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for l-adic intersection complexes.