Acta Mathematica

Enumeration of points, lines, planes, etc.

June Huh and Botong Wang

Full-text: Open access

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


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