General conjecture on general-position subsets of finite affine spaces

Let $d53d_q$ be the finite field with $q$ elements, and let $d53d_q^d$ be its $d$-dimensional affine space. For a $p$-random subset of $d53d_q^d$, let $d6fc(d53d_q^d,p)$ denote the maximum size of a point set in general position, meaning that no $d+1$ points lie in a $(d-1)$-dimensional affine subspace. Here $p$ is the independent probability with which each point is selected.

General conjecture. For every integer $d d45a 2$, there is a positive real number $C=C(d)$ such that, as the prime power $q$ tends to infinity, with high probability,

$$\alpha(\mathbb{F}_q^{d},p)=\left\{\begin{array}{ll}\Theta(pq^d)&\textrm{ for }\quad \omega(q^{-d})=p=o(q^{-d+1/d}),\Theta(pq)&\textrm{ for }\quad Cq^{-1+1/d}\log^2q\leqslant p\leqslant1, \end{array}\right.$$

and

$$\Omega(q^{1/d}/\log q)=\alpha(\mathbb{F}_q^{d},p)=O(q^{1/d}\log^2q)$$

for $q^{-d+1/d}/\log q\leqslant p<Cq^{-1+1/d}\log^2q$, where all implicit constants depend only on $d$. This conjecture predicts the order of magnitude of the largest general-position subset across all relevant ranges of the sampling probability; the paper establishes the three-dimensional case up to polylogarithmic factors, while the general statement remains open.