Journal of Geometry

, Volume 108, Issue 1, pp 33–43 | Cite as

General position subsets and independent hyperplanes in d-space

Article

Abstract

Erdős asked what is the maximum number \({\alpha(n)}\) such that every set of \({n}\) points in the plane with no four on a line contains \({\alpha(n)}\) points in general position. We consider variants of this question for \({d}\)-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed \({d}\):
  • Every set \({\mathcal{H}}\) of \({n}\) hyperplanes in \({\mathbb{R}^d}\) contains a subset \({S\subseteq \mathcal{H}}\) of size at least \({c \left(n \log n\right)^{1/d}}\), for some constant \({c=c(d)> 0}\), such that no cell of the arrangement of \({\mathcal{H}}\) is bounded by hyperplanes of \({S}\) only.

  • Every set of \({cq^d\log q}\) points in \({\mathbb{R}^d}\), for some constant \({c=c(d)> 0}\), contains a subset of \({q}\) cohyperplanar points or \({q}\) points in general position.

Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013].

Mathematics Subject Classification

Primary 52C35 Secondary 52C10 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Csaba D. Tóth
    • 2
  • David R. Wood
    • 3
  1. 1.Université libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.California State University NorthridgeNorthridgeUSA
  3. 3.Monash UniversityMelbourneAustralia

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