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Enumeration of points, lines, planes, etc. (English) Zbl 1386.05021
Summary: 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.

##### MSC:
 05B25 Combinatorial aspects of finite geometries 51E26 Other finite linear geometries
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