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 and Warren (2022) and Bhowmick and Roche-Newton (2024). For d ⩾ 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 2q 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 the result of Roche-Newton and Warren (2022). Our proofs are grounded in the hypergraph container method. In addition, for d = 2, we also leverage the pseudorandomness of the point-line incidence graph of \({\mathbb F}_{q}^{2}\) .
