On the number of lines in 4-dimensional linear spaces (Q1919307)

A four-dimensional linear space is a connected geometry of rank 4 with Buekenhout diagram the string \(L.L.L\). Let \({\mathbf L}\) be a four-dimensional linear space with \(v\) points. Let \(q\) be the unique positive real number such that \(v = q^4 + q^3 + q^2 + q + 1\). The author shows that, if every three-dimensional linear subspace of \({\mathbf L}\) has at least \(v - q^2 - 1\) points, then \({\mathbf L}\) has at least \((q^2 + 1)v\) lines. Moreover, if \({\mathbf L}\) has exactly \((q^2 + 1)v\) lines, then \({\mathbf L}\) is isomorphic to \(PG (4,q)\) (and consequently \(q\) is a prime power). This result generalizes the lower dimensional cases (dimensions 2 and 3). Very likely, the result can be generalized to arbitary dimension.