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June 2017 Enumeration of points, lines, planes, etc.
June Huh, Botong Wang
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Acta Math. 218(2): 297-317 (June 2017). DOI: 10.4310/ACTA.2017.v218.n2.a2

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.

Funding Statement

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

Citation

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June Huh. Botong Wang. "Enumeration of points, lines, planes, etc.." Acta Math. 218 (2) 297 - 317, June 2017. https://doi.org/10.4310/ACTA.2017.v218.n2.a2

Information

Received: 9 October 2016; Revised: 30 January 2017; Published: June 2017
First available in Project Euclid: 31 January 2018

zbMATH: 06826207
MathSciNet: MR3733101
Digital Object Identifier: 10.4310/ACTA.2017.v218.n2.a2

Rights: Copyright © 2017 Institut Mittag-Leffler

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Vol.218 • No. 2 • June 2017
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