On higher dimensional point sets in general position

TL;DR

This work leverages hypergraph containers to obtain new upper bounds on the largest guaranteed general-position subset size $\alpha_d(N)$ for point sets in $\mathbb{R}^d$, proving $\alpha_d(N) < N^{\tfrac{1}{2}+\tfrac{1}{2d}+o(1)}$ for odd $d$ and $\alpha_d(N) < N^{\tfrac{1}{2}+\tfrac{1}{d-1}+o(1)}$ for even $d$, by combining a novel supersaturation lemma for non-degenerate $(k+2)$-tuples on $k$-flats with the container method. The paper also extends these ideas to generalized parameters $\alpha_{d,s}(N)$ and to grid-based extremal functions $a(d,k,n)$, obtaining improved polynomial bounds when $k+2\equiv 0$ or $1 \pmod{4}$, including the explicit improvement $a(4,2,n)\le O(n^{16/9})$ over Lefmann’s bound. Through a detailed hypergraph framework and probabilistic deletions, the authors showcase how container methods yield nontrivial bounds in high-dimensional geometric incidence problems and discuss related lower bounds, open questions, and connections to $B_g$-sets and multilevel generalizations.

Abstract

A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $α_d(N)$ 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 $α_d(N)$ in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that $α_2(N) < N^{5/6 + o(1)}$. In this paper, we also use the container method to obtain new upper bounds for $α_d(N)$ when $d \geq 3$. More precisely, we show that if $d$ is odd, then $α_d(N) < N^{\frac{1}{2} + \frac{1}{2d} + o(1)}$, and if $d$ is even, we have $α_d(N) < N^{\frac{1}{2} + \frac{1}{d-1} + o(1)}$. 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).

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Claim 2.2
• proof : Proof of Claim
• Claim 2.3
• proof : Proof of Claim
• Lemma 2.4