A 4-general set in \(\textrm{PG}(n, q)\) is a set of points of \(\textrm{PG}(n, q)\) spanning the whole \(\textrm{PG}(n, q)\) 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 \(\textrm{PG}(n, q)\) . In this paper, upper and lower bounds for the size of the largest and the smallest complete 4-general set in \(\textrm{PG}(n, q)\) , respectively, are investigated. Complete 4-general sets in \(\textrm{PG}(n, q)\) , \(q \in \{3, 4\}\) , 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 \(\textrm{PG}(5, q)\) , \(q \equiv 1 \pmod {3}\) .
