Maximum number of points in general position in a random subset of finite $3$-dimensional spaces

TL;DR

The paper addresses the problem of determining the maximum size $\alpha(\mathbb{F}_q^{3},p)$ of a general-position subset within a $p$-random subset of $\mathbb{F}_q^3$. It develops a balanced supersaturation result for $d=3$ by analyzing degeneracy patterns and constructing a random 4-sets hypergraph $\mathcal{H}_U$ with controlled co-degrees, using Chernoff bounds and the Hypergraph Container Method. This leads to the bound $|\mathcal{H}_U| = \Omega(n^{4}/q)$ and $\Delta_i(\mathcal{H}_U) = O(n^{4-i} / q^{1 - (i-1)/3})$ for $i=1,2,3$, which in turn, via the Hypergraph Container Method, yields the order of magnitude of $\alpha(\mathbb{F}_q^{3},p)$ up to polylog factors. The concluding discussion highlights that the $d=3$ case is resolved with this approach, while higher dimensions pose substantial obstacles, motivating further research.

Abstract

Let $α(\mathbb{F}_q^{d},p)$ be the maximum possible size of a point set in general position in the $p$-random subset of $\mathbb{F}_q^d$. In this note, we determine the order of magnitude of $α(\mathbb{F}_q^{3},p)$ up to a polylogarithmic factor by proving a balanced supersaturation result for the sets of $4$ points in the same plane.

Figures (1)

• Figure 1: The behavior of $\alpha(\mathbb{F}_q^{3},p)$ in terms of $p$.

Theorems & Definitions (20)

• Conjecture 1
• Conjecture 2: Balanced supersaturation
• Claim 3
• Theorem 4
• Lemma 5
• proof : Proof of \ref{['3dBalSup']}
• Claim 6
• proof
• Claim 7
• proof