Random Turán and counting results for general position sets over finite fields

TL;DR

The paper investigates extremal questions for general position sets in finite field spaces $\mathbb{F}_q^d$, focusing on (i) the maximum size $\alpha(\mathbb{F}_q^d,p)$ of a general position subset within a random subset, and (ii) the total count of general position sets in $\mathbb{F}_q^d$. Central to the approach is the hypergraph container method, applied to the coplanarity hypergraph $\mathcal{H}_{q,d}$ and its variants, together with incidence-structure pseudorandomness in low dimensions. The authors obtain: (a) sharp or near-sharp bounds for $\alpha(\mathbb{F}_q^2,p)$ across all $p$ and essentially tight upper bounds for $\alpha(\mathbb{F}_q^d,p)$ in higher dimensions within certain $p$-ranges; (b) asymptotically tight (in the exponent) upper bounds for the total number of general position sets in $\mathbb{F}_q^d$ and, for fixed sizes $m$, near-optimal bounds extending prior work to all $d$ (with a dimension-2 improvement); and (c) refined counting results leveraging pseudorandomness of the 2D point-line incidence graph. The results deepen the understanding of random Turán-type phenomena and counting in finite-field geometries, with techniques adaptable to higher-dimensional incidence and evasive-set problems. Overall, the work advances the exact and near-exact characterization of general-position phenomena in finite field geometries via containers and supersaturation arguments.

Abstract

Let $α(\mathbb{F}_q^d,p)$ denote the maximum size of a general position set in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $α(\mathbb{F}_q^2,p)$ up to polylogarithmic factors for all possible values of $p$, improving the previous results obtained by Roche-Newton--Warren and Bhowmick--Roche-Newton. For $d \ge 3$ we prove upper bounds for $α(\mathbb{F}_q^d,p)$ that are essentially tight within certain ranges for $p$. We establish the upper bound $2^{(1+o(1))q}$ for the number of general position sets in $\mathbb{F}_q^d$, which matches the trivial lower bound $2^{q}$ asymptotically in the exponent. We also refine this counting result by proving an asymptotically tight (in the exponent) upper bound for the number of general position sets with a fixed size. The latter result for $d=2$ improves a result of Roche-Newton--Warren. Our proofs are grounded in the hypergraph container method, and additionally, for $d=2$ we also leverage the pseudorandomness of the point-line incidence graph of $\mathbb{F}_{q}^2$.

Figures (2)

• Figure 1: The behaviour of $\alpha(\mathbb{F}_q^2,p)$ in terms of $p$.
• Figure 2: Lower and upper bounds for $\alpha(\mathbb{F}_q^d,p)$ in terms of $p$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: Goldman--Rota goldman1969number
• Lemma 2.2: Balogh--Morris--Samotij balogh2015independent, Saxton--Thomason saxton2015hypergraph
• Lemma 3.1
• Lemma 3.2
• proof