Enumeration of points, lines, planes, etc.

*(English)*Zbl 1386.05021Summary: One of the earliest results in enumerative combinatorial geometry is the following theorem of N. G. de Bruijn and P. Erdős [Proc. Akad. Wet. Amsterdam 51, 1277–1279 (1948; Zbl 0032.24405)]: 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 \(\frac{1}{2} r\). 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 \(\ell\)-adic intersection complexes.