Letα(F_(q)^(d),p)denote the maximum size of a general position set in a p-random subset of F_(q)^(d).We determine the order of magnitude ofα(F_(q)^(2),p)up to polylogarithmic factors for all possible values of p,imp...Letα(F_(q)^(d),p)denote the maximum size of a general position set in a p-random subset of F_(q)^(d).We determine the order of magnitude ofα(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 forα(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 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 F_(q)^(2).展开更多
基金supported by European Research Council Advanced Grant(Grant No.101020255)Leverhulme Research Project Grant(Grant No.RPG-2018-424)+3 种基金supported by National Natural Science Foundation of China(Grant No.123B2012)supported by European Research Council Advanced Grants“GeoScape”(Grant No.882971)and“ERMiD”(Grant No.101054936)Jonathan Tidor for stimulating discussions at 2023 University of California San Diego Workshop on Ramsey Theoryinitiated while Ji Zeng was visiting Shanghai Center for Mathematical Sciences at the kind invitation of Hehui Wu.
文摘Letα(F_(q)^(d),p)denote the maximum size of a general position set in a p-random subset of F_(q)^(d).We determine the order of magnitude ofα(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 forα(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 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 F_(q)^(2).
