Sets of type \((q-1,n)\) in \(PG(3, q)\) (Q6601251)

In this paper, the authors prove the following interesting and strong result. Consider \(\mathbf{P} = PG(3, q)\), the projective geometry of dimension \(3\) over \(\mathbb{F}_q\). A set \(K\) of points of \(\mathbf{P}\) is said to be type \((m, n)\) (with \(m < n\)) if \(K\) intersects every plane in \(\mathbf{P}\) in \(m\) or \(n\) points. \(K\) is then called an \((m, n)\)-set. Let \(K\) be an \((m, n)\)-set in \(\mathbf{P}\). In an earlier result of Tallini Scafati, it was shown that \(n - m\) must divide \(q^2\). Let \(q = p^h\). In an earlier paper, the authors proved that if \(p\) is the characteristic of \(\mathbb{F}_q\), then \(p \leq n - m \leq q\). Use of these two earlier results and a lot of computation (mainly involving careful two-way counting), the present paper proves the following strong result: If \(K\) is an \((m, n)\)-set in \(\mathbf{P}\), with \(m = q - 1\) then \(n = 2q - 1 = m + q\).