On 4-general sets in finite projective spaces

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