Linear spaces with few lines (Q5905661)

Author considers the following problem. What are finite linear spaces (not being a near-pencil) with possibly small number of lines; are they all parts of a projective plane. There is known a lower bound \(B(\nu)\) for number \(b\) of lines of a linear space with \(\nu\) points defined (by Erdős) as follows. Consider a (unique) positive integer \(n\) with \(n^ 2 - n+2 \leq \nu \leq n^ 2 + n + 1\); then either \(\nu = n^ 2 - n + 2\), or \(n^ 2 - n + 3 \leq \nu \leq n^ 2 + 1\), or \(n^ 2 + 2 \leq \nu\). In accordance, define \(B(\nu)\) to be \(n^ 2 + n-1\), \(n^ 2 + n\), or \(n^ 2 + n + 1\), resp.; then \(B(\nu) \leq b\). The author is interested in structures for which \(b = B(\nu)\) holds and he proves that every such space can be embedded into a projective space of order \(n\), except exactly one (defined in the literature) \(\mathbb{E}_ 1\) with 8 points. Next the author considers structures for which \(b = B(\nu)+1\) holds. The cases \(n^ 2 - n + 3 \leq \nu \leq n^ 2+1\) and \(n^ 2 + 2\leq \nu\) were discussed in other papers of the author. In this paper it is shown that if \(\nu = n^ 2 - n+2\) and \(b = B(\nu) + 1\), then the corresponding space can be embedded into a projective plane, or it is another exceptional structure \(\mathbb{E}_ 2\) with 8 points.