## Acta Mathematica

- Acta Math.
- Volume 218, Number 2 (2017), 297-317.

### Enumeration of points, lines, planes, etc.

June Huh and Botong Wang

#### Abstract

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 $\lvert E \rvert$ 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 $\lvert E \rvert$ 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.

#### Note

June Huh was supported by a Clay Research Fellowship and NSF Grant DMS-1128155.

#### Article information

**Source**

Acta Math., Volume 218, Number 2 (2017), 297-317.

**Dates**

Received: 9 October 2016

Revised: 30 January 2017

First available in Project Euclid: 31 January 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.acta/1517426685

**Digital Object Identifier**

doi:10.4310/ACTA.2017.v218.n2.a2

**Mathematical Reviews number (MathSciNet)**

MR3733101

**Zentralblatt MATH identifier**

06826207

#### Citation

Huh, June; Wang, Botong. Enumeration of points, lines, planes, etc. Acta Math. 218 (2017), no. 2, 297--317. doi:10.4310/ACTA.2017.v218.n2.a2. https://projecteuclid.org/euclid.acta/1517426685