On 4-general sets in finite projective spaces

05/23/2023

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by   Francesco Pavese, et al.

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A 4-general set in PG(n,q) is a set of points of PG(n,q) spanning the whole PG(n,q) and such that no four of them are on a plane. Such a pointset is said to be complete if it is not contained in a larger 4-general set of PG(n, q). In this paper upper and lower bounds for the size of the largest and the smallest complete 4-general set in PG(n,q), respectively, are investigated. Complete 4-general sets in PG(n,q), q ∈{3,4}, whose size is close to the theoretical upper bound are provided. Further results are also presented, including a description of the complete 4-general sets in projective spaces of small dimension over small fields and the construction of a transitive 4-general set of size 3(q + 1) in PG(5, q), q ≡ 1 3.