A characterization of finite linear spaces on \(v\) points, \(n^ 2\leqslant v < (n+1)^ 2\), and \(b = n^ 2+n+3\) lines, \(n > 10\) (Q685597)

Let \({\mathcal S} = (P, L)\) be a (finite) linear space with \(\nu\) points and \(b\) lines (then \(\nu \leq b\)); there is the unique integer \(n\) with \(n^ 2 \leq \nu (n + 1)^ 2\). Various results relating \(n\) and \(b\) are known -- in the article linear spaces with \(b = n^ 2 + n + 3\) (\(n\geq 10\)) are characterized. So \(\mathcal S\), if not a near-pencil, must be an affine plane of order \(n\) with \(q\) \((0 \leq q \leq 3\)) points deleted, enlarged by three additional points, joined by three new additional lines in a triangle. The proof goes as follows: if \(n^ 2 + 3 \leq \nu\), the theorem follows by known results. Thus \(\nu = n^ 2\), or \(\nu = n^ 2 + 1\), or \(\nu = n^ 2 + 2\). In each of these three cases, direct (though long) combinatorial calculations yield the result.