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 \({d}\):
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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.
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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].
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Research of Wood is supported by the Australian Research Council.
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Cardinal, J., Tóth, C.D. & Wood, D.R. General position subsets and independent hyperplanes in d-space. J. Geom. 108, 33–43 (2017). https://doi.org/10.1007/s00022-016-0323-5
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DOI: https://doi.org/10.1007/s00022-016-0323-5