Let α(Fqd​,p) denote the maximum size of a general position
set in a p-random subset of Fqd​. We determine the order of
magnitude of α(Fq2​,p) up to polylogarithmic factors for all
possible values of p, improving the previous best upper bounds obtained by
Roche-Newton--Warren and Bhowmick--Roche-Newton. For d≥3 we prove upper
bounds for α(Fqd​,p) that are essentially tight within certain
intervals of p.
We establish the upper bound 2(1+o(1))q for the number of general
position sets in Fqd​, which matches the trivial lower bound
2q asymptotically in exponent. We also refine this counting result by
proving an asymptotically tight (in exponent) upper bound for the number of
general position sets with 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
bipartite graph of Fq2​.Comment: 24 pages(+2 pages for Appendix), 2 figure