The Minimum Size of a Maximal Partial Plane

A partial plane of order \(n\) is a family \(\mathcal{L}\) of \(n+1\)-element subsets of an \(n^2+n+1\)-element set, such that no two sets meet more than \(1\) element. Here it is proved, that if \(\mathcal{L}\) is maximal, then \(|\mathcal{L}| \geq \lfloor\frac{3n}{2}\rfloor + 2\), and this inequality is sharp.