On higher dimensional point sets in general position

TL;DR

This paper improves bounds on the size of large point sets in general position in higher dimensions using hypergraph container methods, and enhances known results on grid point configurations avoiding certain flat alignments.

Contribution

It extends the hypergraph container approach to higher dimensions for bounding point sets in general position and refines bounds for grid point configurations avoiding specific flat alignments.

Findings

New upper bounds for (N) when d  3.

Improved bounds for a(d,k,n) for k=2 or 3 (mod 4).

Application of hypergraph containers to geometric combinatorics.

Abstract

A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $\alpha_d(N)$ be the largest integer such that any set of $N$ points in $\mathbb{R}^d$, with no $d + 2$ members on a common hyperplane, contains a subset of size $\alpha_d(N)$ in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that $\alpha_2(N) < N^{5/6 + o(1)}$. In this paper, we also use the container method to obtain new upper bounds for $\alpha_d(N)$ when $d \geq 3$. More precisely, we show that if $d$ is odd, then $\alpha_d(N) < N^{\frac{1}{2} + \frac{1}{2d} + o(1)}$, and if $d$ is even, we have $\alpha_d(N) < N^{\frac{1}{2} + \frac{1}{d-1} + o(1)}$. We also study the classical problem of determining $a(d,k,n)$, the maximum number of points selected from the grid $[n]^d$ such that no $k + 2$ members lie on a $k$-flat, and improve the previously best known bound for $a(d,k,n)$, due to Lefmann in 2008, by a polynomial factor when $k$ = 2 or 3 (mod 4).