The nonexistence of finite linear spaces with \(v=n^ 2\) points and \(b=n^ 2 + n + 2\) lines (Q1801681)

Let \(S\) be a finite linear space with \(v\) points and \(b\) lines and with \(n^ 2\leq v\leq b=n^ 2+n+2\) for some integer \(n\). If \(v>n^ 2\), then \textit{D. R. Stinson} [Geom. Dedicata 13, 429-434 (1989; Zbl 0523.51013)] proves that \(n\leq 3\), and all cases are described. In the paper under review the author considers the case \(v=n^ 2\) and shows that necessarily \(n\leq 4\) and all possibilities are determined and mentioned, but not defined nor described. The proof of the result uses only elementary counting techniques together with theorems of de Witte, Stinson and Metsch.