General Position Subsets and Independent Hyperplanes in d-Space

TL;DR

This paper extends classical geometric combinatorics questions to higher dimensions, establishing bounds on large subsets of points or hyperplanes with specific properties in d-dimensional space.

Contribution

It generalizes known 2D results to d-dimensions, providing new bounds on subsets of hyperplanes and points with particular geometric configurations.

Findings

Existence of large hyperplane subsets with unbounded cells

Presence of large cohyperplanar or general position point subsets

Extension of 2D geometric bounds to higher dimensions

Abstract

Erd\H{o}s 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 $H$ of $n$ hyperplanes in $\mathbb{R}^d$ contains a subset $S\subseteq H$ 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 $H$ 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].