General-dimensional random general-position set conjecture

Determine the asymptotic order of the maximum size α(F_q^d,p) of a general-position subset of a p-random subset of the d-dimensional affine space over the finite field F_q for every integer d>2 and all ranges of p specified in Conjecture 1, with high probability as q tends to infinity.

Background

The paper studies α(F_qd,p), the largest size of a point set in general position contained in a random subset of F_qd. Prior work establishes essentially sharp results in dimension two and in selected parameter ranges in higher dimensions. Conjecture 1 proposes a complete order-of-magnitude description across all dimensions d>2 and all relevant probabilities p.

The paper proves the conjecture only up to a polylogarithmic factor in dimension three, conditional on and via a balanced supersaturation result. The general-dimensional statement therefore remains unresolved, except in the cases and ranges covered by earlier results and the paper’s deductions.

References

For the general case, the following conjecture is natural, based on the previous results mentioned above. It was also implicitly mentioned in Section 6 in [6]. Conjecture 1. For every integer d > 2, there is a positive real number C = C(d) such that the following is true.

— Maximum number of points in general position in a random subset of finite $3$-dimensional spaces  (2503.04102 - Balogh et al., 6 Mar 2025) in Conjecture 1, Section 1, p. 2