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 \1/2 r. This confirms the “top-heavy” conjecture by Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for \-adic intersection complexes.