Sets of type \((q,n)\) in \(\mathrm{PG}(3,q)\) (Q310485)

A proper non-empty subset \(K\) of \(\mathrm{PG}(3,q)\) is of type \((m,n)\) if the integers \(|\pi \cap K|\), where \(\pi\) runs over the planes of \(\mathrm{PG}(3,q),\) are the set \(\{m,n\}\). The authors prove that for subsets \(K\) of type \((q,n)\), if \(q\) is a prime or \(K\) contains a line then \(n=2q\). Moreover, they compute that if \(K\) does not contain a line and \(q \leq 10^9\), then \(n=2q\).