## Abstract

Erdős asked what is the maximum number (Formula presented.) such that every set of (Formula presented.) points in the plane with no four on a line contains (Formula presented.) points in general position. We consider variants of this question for (Formula presented.)-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed (Formula presented.):Every set (Formula presented.) of (Formula presented.) hyperplanes in (Formula presented.) contains a subset (Formula presented.) of size at least (Formula presented.), for some constant (Formula presented.), such that no cell of the arrangement of (Formula presented.) is bounded by hyperplanes of (Formula presented.) only.Every set of (Formula presented.) points in (Formula presented.), for some constant (Formula presented.), contains a subset of (Formula presented.) cohyperplanar points or (Formula presented.) 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].

Original language | English |
---|---|

Pages (from-to) | 33-43 |

Number of pages | 11 |

Journal | Journal of Geometry |

Volume | 108 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |