Subsets of PG\((n,2)\) and maximal partial spreads in PG\((4,2)\) (Q5926325)

Let \(\theta_n\) be the number of points in \(PG(n,2)\) and let \(\phi_n\) be the number of lines in \(PG(n,2)\). Let \(\Psi\) be any point-subset of \(PG(n,2)\). Let \(L\) be the number of internal lines of \(\Psi\) and let \(L'\) be the number of external lines of \(\Psi\). The author shows that the sum \(L+L'\) is the same for all \(\Psi\) having the same cardinality. In particular, if \(k=|\Psi|-\theta_{n-1}\), then \(L+L'=\phi_{n-1}+k(k-1)/2\). Let \(\mathcal S\) be a partial spread of lines in \(PG(4,2)\), and let \(N\) denote the number of reguli contained in \(\mathcal S\). Using the above result, the author gives a simple proof of the following theorem: if \(\mathcal S\) is maximal then either \(|{\mathcal S}|=5\) and \(N=10\), or \(|{\mathcal S}|=7\) and \(N=4\), or \(|{\mathcal S}|=9\) and \(N=4\). He also furnishes a detailed description of these partial spreads.