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

翻译：在${\rm PG}(n,q)$中，一个$4$-一般集合是指张成整个${\rm PG}(n,q)$且其中任意四点不共面的点集。若这样的点集不包含于${\rm PG}(n,q)$中更大的$4$-一般集合，则称其为完全的。本文研究了${\rm PG}(n,q)$中最大和最小完全$4$-一般集合的大小分别对应的上界和下界。我们给出了${\rm PG}(n,q)$（$q \in \{3,4\}$）中大小接近理论上限的完全$4$-一般集合构造。此外，还提供了进一步结果，包括对小域上低维射影空间中完全$4$-一般集合的描述，以及在${\rm PG}(5, q)$（$q \equiv 1 \pmod{3}$）中大小为$3(q + 1)$的可递$4$-一般集合的构造。