Random Tur\'an and counting results for general position sets over finite fields

TL;DR

This paper investigates the size and count of general position sets in finite field vector spaces, providing new bounds and asymptotic results using hypergraph containers and pseudorandomness techniques.

Contribution

It improves bounds on the maximum size of random general position sets and establishes tight asymptotic counts for these sets in finite fields.

Findings

Determined the order of magnitude of $oldsymbol{ ext{alpha}}(oldsymbol{ ext{F}}_q^2,p)$ for all $p$.

Proved essentially tight upper bounds for $oldsymbol{ ext{alpha}}(oldsymbol{ ext{F}}_q^d,p)$ for $d \\ge 3$.

Established an asymptotically tight upper bound for the number of general position sets in $oldsymbol{ ext{F}}_q^d$.

Abstract

Let $\alpha(\mathbb{F}_q^d,p)$ 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 $\alpha(\mathbb{F}_q^2,p)$ 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 $\alpha(\mathbb{F}_q^d,p)$ 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 $\mathbb{F}_{q}^2$.